{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261106-r06-linear-diophantine-equation-0064",
  "pilot_number": "R0038-19639",
  "number": 19639,
  "family_id": "qf-mod37-dioph-general-solution",
  "title": {
    "zh-Hant": "第 19639 題：線性Diophantine方程",
    "en": "Question 19639: All Solutions Need the Gcd-Normalized Integer Parameter"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261106-r06-linear-diophantine-equation-0064",
    "batch_id": "batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-diophantine-equation"
    ],
    "title_zh": "線性Diophantine方程",
    "stem_zh": "exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-90,\"b\":135,\"c\":630}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)",
      "explanation_zh": "Wave5封閉層核對=(14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "線性Diophantine方程"
      ]
    },
    "variant": {
      "family_id": "qf-mod37-dioph-general-solution",
      "seed": 202611066064,
      "parameters": {
        "a": -90,
        "b": 135,
        "c": 630,
        "domain": "Exact_integer_linear_Diophantine_equation",
        "operation": "dioph-general-solution",
        "semantic_template": "qf-mod37-dioph-general-solution/{\"a\":-90,\"b\":135,\"c\":630}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261106-r06-linear-diophantine-equation-0064",
    "batch_id": "batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "cbca5033bd88cf4483e307e6762415892009d6fead99a75cba8b6bccf0a1da0e",
      "family_id": "qf-mod37-dioph-general-solution",
      "seed": 202611066064,
      "semantic_template": "qf-mod37-dioph-general-solution/{\"a\":-90,\"b\":135,\"c\":630}"
    },
    "fields": {
      "title": "All Solutions Need the Gcd-Normalized Integer Parameter",
      "stem": "Use exact original integers a=-90, b=135, c=630 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)",
        "explanation": "For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)."
      },
      "skills": [
        "All Solutions Need the Gcd-Normalized Integer Parameter"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-90,\"b\":135,\"c\":630}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use exact original integers a=-90, b=135, c=630 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=dioph-general-solution;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[x0, y0, step_x, step_y, x_formula, y_formula, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不自動成為數學證據，No依各題聲稱範圍解讀。",
        "en": "Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "ORIGINAL signed coefficients a=-90,b=135",
      "after": "start exact Euclidean chain on absolute values=[90, 135];nonnegative gcd;source deterministic coefficient recurrence",
      "relation": "implies",
      "rule_id": "dioph_original_signs_deterministic_extended_chain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gcd以非負慣例計在abs(a),abs(b)，但Bézout最後要還原原符號。來源固定這條extended chain的係數，不能換另一組數值正確卻不同的答案。",
        "en": "Use absolute inputs for the nonnegative gcd,but restore original signs in Bezout coefficients.Retain this source deterministic extended-chain pair rather than substituting another valid but different pair."
      },
      "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": "Dioph Euclid step0 exact ratio := div(90, 135)",
      "after": "Dioph Euclid step0 exact ratio = 2/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step0 exact ratio，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step0 exact ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "90",
          "135"
        ],
        "output": "2/3",
        "label": "Dioph Euclid step0 exact ratio"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Dioph Euclid step0 divisor multiple := mul(0, 135)",
      "after": "Dioph Euclid step0 divisor multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step0 divisor multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step0 divisor multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "135"
        ],
        "output": "0",
        "label": "Dioph Euclid step0 divisor multiple"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Dioph Euclid step0 remainder := sub(90, 0)",
      "after": "Dioph Euclid step0 remainder = 90",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step0 remainder，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step0 remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "90",
          "0"
        ],
        "output": "90",
        "label": "Dioph Euclid step0 remainder"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Dioph Euclid step0 first coefficient multiple := mul(0, 0)",
      "after": "Dioph Euclid step0 first coefficient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step0 first coefficient multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step0 first coefficient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Dioph Euclid step0 first coefficient multiple"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Dioph Euclid step0 first coefficient difference := sub(1, 0)",
      "after": "Dioph Euclid step0 first coefficient difference = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step0 first coefficient difference，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step0 first coefficient difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "Dioph Euclid step0 first coefficient difference"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Dioph Euclid step0 second coefficient multiple := mul(0, 1)",
      "after": "Dioph Euclid step0 second coefficient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step0 second coefficient multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step0 second coefficient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Dioph Euclid step0 second coefficient multiple"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Dioph Euclid step0 second coefficient difference := sub(0, 0)",
      "after": "Dioph Euclid step0 second coefficient difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step0 second coefficient difference，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step0 second coefficient difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Dioph Euclid step0 second coefficient difference"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "step=0;positive divisor=135;old pair=[90, 135];exact ratio=2/3;floor quotient=0;old coefficient pairs=[[1, 0], [0, 1]]",
      "after": "next remainder=90 in[0,135);next signed absolute-input coefficients=[1, 0];common divisor set unchanged",
      "relation": "implies",
      "rule_id": "dioph_every_extended_step_invariant_and_descent",
      "conditions": [],
      "explanation": {
        "zh-Hant": "left−q·right保與right同一公因數集，同一q更新兩個係數保abs(a)·s+abs(b)·t等餘數。下一餘數非負且嚴格小於正divisor，故終止，且從不除以0。",
        "en": "Remainder subtraction preserves the common-divisor set,and the same quotient updates both coefficients so abs(a)*s+abs(b)*t equals the remainder.Nonnegative strict remainder descent guarantees termination without division by0."
      },
      "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": "Dioph Euclid step1 exact ratio := div(135, 90)",
      "after": "Dioph Euclid step1 exact ratio = 3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step1 exact ratio，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step1 exact ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "135",
          "90"
        ],
        "output": "3/2",
        "label": "Dioph Euclid step1 exact ratio"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Dioph Euclid step1 divisor multiple := mul(1, 90)",
      "after": "Dioph Euclid step1 divisor multiple = 90",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step1 divisor multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step1 divisor multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "90"
        ],
        "output": "90",
        "label": "Dioph Euclid step1 divisor multiple"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Dioph Euclid step1 remainder := sub(135, 90)",
      "after": "Dioph Euclid step1 remainder = 45",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step1 remainder，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step1 remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "135",
          "90"
        ],
        "output": "45",
        "label": "Dioph Euclid step1 remainder"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Dioph Euclid step1 first coefficient multiple := mul(1, 1)",
      "after": "Dioph Euclid step1 first coefficient multiple = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step1 first coefficient multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step1 first coefficient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Dioph Euclid step1 first coefficient multiple"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Dioph Euclid step1 first coefficient difference := sub(0, 1)",
      "after": "Dioph Euclid step1 first coefficient difference = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step1 first coefficient difference，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step1 first coefficient difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "1"
        ],
        "output": "-1",
        "label": "Dioph Euclid step1 first coefficient difference"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Dioph Euclid step1 second coefficient multiple := mul(1, 0)",
      "after": "Dioph Euclid step1 second coefficient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step1 second coefficient multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step1 second coefficient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Dioph Euclid step1 second coefficient multiple"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Dioph Euclid step1 second coefficient difference := sub(1, 0)",
      "after": "Dioph Euclid step1 second coefficient difference = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step1 second coefficient difference，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step1 second coefficient difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "Dioph Euclid step1 second coefficient difference"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "step=1;positive divisor=90;old pair=[135, 90];exact ratio=3/2;floor quotient=1;old coefficient pairs=[[0, 1], [1, 0]]",
      "after": "next remainder=45 in[0,90);next signed absolute-input coefficients=[-1, 1];common divisor set unchanged",
      "relation": "implies",
      "rule_id": "dioph_every_extended_step_invariant_and_descent",
      "conditions": [],
      "explanation": {
        "zh-Hant": "left−q·right保與right同一公因數集，同一q更新兩個係數保abs(a)·s+abs(b)·t等餘數。下一餘數非負且嚴格小於正divisor，故終止，且從不除以0。",
        "en": "Remainder subtraction preserves the common-divisor set,and the same quotient updates both coefficients so abs(a)*s+abs(b)*t equals the remainder.Nonnegative strict remainder descent guarantees termination without division by0."
      },
      "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": "Dioph Euclid step2 exact ratio := div(90, 45)",
      "after": "Dioph Euclid step2 exact ratio = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step2 exact ratio，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step2 exact ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "90",
          "45"
        ],
        "output": "2",
        "label": "Dioph Euclid step2 exact ratio"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Dioph Euclid step2 divisor multiple := mul(2, 45)",
      "after": "Dioph Euclid step2 divisor multiple = 90",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step2 divisor multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step2 divisor multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "45"
        ],
        "output": "90",
        "label": "Dioph Euclid step2 divisor multiple"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Dioph Euclid step2 remainder := sub(90, 90)",
      "after": "Dioph Euclid step2 remainder = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step2 remainder，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step2 remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "90",
          "90"
        ],
        "output": "0",
        "label": "Dioph Euclid step2 remainder"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Dioph Euclid step2 first coefficient multiple := mul(2, -1)",
      "after": "Dioph Euclid step2 first coefficient multiple = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step2 first coefficient multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step2 first coefficient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "-2",
        "label": "Dioph Euclid step2 first coefficient multiple"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Dioph Euclid step2 first coefficient difference := sub(1, -2)",
      "after": "Dioph Euclid step2 first coefficient difference = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step2 first coefficient difference，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step2 first coefficient difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "3",
        "label": "Dioph Euclid step2 first coefficient difference"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Dioph Euclid step2 second coefficient multiple := mul(2, 1)",
      "after": "Dioph Euclid step2 second coefficient multiple = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step2 second coefficient multiple，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step2 second coefficient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Dioph Euclid step2 second coefficient multiple"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Dioph Euclid step2 second coefficient difference := sub(0, 2)",
      "after": "Dioph Euclid step2 second coefficient difference = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph Euclid step2 second coefficient difference，不以小數近似替代。",
          "en": "Recompute Dioph Euclid step2 second coefficient difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "2"
        ],
        "output": "-2",
        "label": "Dioph Euclid step2 second coefficient difference"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "step=2;positive divisor=45;old pair=[90, 45];exact ratio=2;floor quotient=2;old coefficient pairs=[[1, 0], [-1, 1]]",
      "after": "next remainder=0 in[0,45);next signed absolute-input coefficients=[3, -2];common divisor set unchanged",
      "relation": "implies",
      "rule_id": "dioph_every_extended_step_invariant_and_descent",
      "conditions": [],
      "explanation": {
        "zh-Hant": "left−q·right保與right同一公因數集，同一q更新兩個係數保abs(a)·s+abs(b)·t等餘數。下一餘數非負且嚴格小於正divisor，故終止，且從不除以0。",
        "en": "Remainder subtraction preserves the common-divisor set,and the same quotient updates both coefficients so abs(a)*s+abs(b)*t equals the remainder.Nonnegative strict remainder descent guarantees termination without division by0."
      },
      "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": "Dioph restore original a sign := mul(-1, -1)",
      "after": "Dioph restore original a sign = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph restore original a sign，不以小數近似替代。",
          "en": "Recompute Dioph restore original a sign with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "Dioph restore original a sign"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Dioph restore original b sign := mul(1, 1)",
      "after": "Dioph restore original b sign = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph restore original b sign，不以小數近似替代。",
          "en": "Recompute Dioph restore original b sign with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Dioph restore original b sign"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Dioph ORIGINAL a times Bezout x := mul(-90, 1)",
      "after": "Dioph ORIGINAL a times Bezout x = -90",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph ORIGINAL a times Bezout x，不以小數近似替代。",
          "en": "Recompute Dioph ORIGINAL a times Bezout x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-90",
          "1"
        ],
        "output": "-90",
        "label": "Dioph ORIGINAL a times Bezout x"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Dioph ORIGINAL b times Bezout y := mul(135, 1)",
      "after": "Dioph ORIGINAL b times Bezout y = 135",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph ORIGINAL b times Bezout y，不以小數近似替代。",
          "en": "Recompute Dioph ORIGINAL b times Bezout y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "135",
          "1"
        ],
        "output": "135",
        "label": "Dioph ORIGINAL b times Bezout y"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Dioph original signed Bezout identity := sum(-90, 135)",
      "after": "Dioph original signed Bezout identity = 45",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph original signed Bezout identity，不以小數近似替代。",
          "en": "Recompute Dioph original signed Bezout identity with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-90",
          "135"
        ],
        "output": "45",
        "label": "Dioph original signed Bezout identity"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "complete exact chain=[[90, 135, 0, 90], [135, 90, 1, 45], [90, 45, 2, 0]];terminal pair=[45, 0];sign-restored x=1,y=1",
      "after": "a*x+b*y=45=nonnegative gcd d=45;gcd from invariance and terminal gcd(d,0)=d",
      "relation": "implies",
      "rule_id": "dioph_terminal_gcd_and_original_signed_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "最後一個非零餘數是gcd，原符號代回再核a*x+b*y=d。a或b一個為0也保同一邊界，不新增除以0的一列。",
        "en": "The last nonzero remainder is the gcd;substitute the original signs to verify a*x+b*y=d.One zero coefficient is legitimate and creates no extra division-by-zero row."
      },
      "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": "Dioph target divided by positive gcd := div(630, 45)",
      "after": "Dioph target divided by positive gcd = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph target divided by positive gcd，不以小數近似替代。",
          "en": "Recompute Dioph target divided by positive gcd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "630",
          "45"
        ],
        "output": "14",
        "label": "Dioph target divided by positive gcd"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "ACTUAL original equation-90*x+135*y=630;positive gcd=45;exact c/d=14;Bezout coefficients=[1, 1]",
      "after": "d divides both coefficients,so ANY integer solution requires d divides c;conversely integer c/d scales Bezout to a solution;solvable=是",
      "relation": "implies",
      "rule_id": "dioph_gcd_divisibility_iff_not_finite_search",
      "conditions": [],
      "explanation": {
        "zh-Hant": "必要性由gcd整除任意整數線性組合；充分性由已給Bézout等式乘c/d。非整數商是全體整數解的障礙，不是搜尋有限範圍沒找到。",
        "en": "The gcd divides every integer linear combination,giving necessity.An integer c/d scales the displayed Bezout identity,giving sufficiency.A noninteger quotient obstructs all integer solutions,not merely a finite search range."
      },
      "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": "Dioph particular x0 from scaled source Bezout := mul(1, 14)",
      "after": "Dioph particular x0 from scaled source Bezout = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph particular x0 from scaled source Bezout，不以小數近似替代。",
          "en": "Recompute Dioph particular x0 from scaled source Bezout with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "14"
        ],
        "output": "14",
        "label": "Dioph particular x0 from scaled source Bezout"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Dioph particular y0 from scaled source Bezout := mul(1, 14)",
      "after": "Dioph particular y0 from scaled source Bezout = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph particular y0 from scaled source Bezout，不以小數近似替代。",
          "en": "Recompute Dioph particular y0 from scaled source Bezout with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "14"
        ],
        "output": "14",
        "label": "Dioph particular y0 from scaled source Bezout"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Dioph original a times particular x0 := mul(-90, 14)",
      "after": "Dioph original a times particular x0 = -1260",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph original a times particular x0，不以小數近似替代。",
          "en": "Recompute Dioph original a times particular x0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-90",
          "14"
        ],
        "output": "-1260",
        "label": "Dioph original a times particular x0"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Dioph original b times particular y0 := mul(135, 14)",
      "after": "Dioph original b times particular y0 = 1890",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph original b times particular y0，不以小數近似替代。",
          "en": "Recompute Dioph original b times particular y0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "135",
          "14"
        ],
        "output": "1890",
        "label": "Dioph original b times particular y0"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Dioph particular original equation check := sum(-1260, 1890)",
      "after": "Dioph particular original equation check = 630",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph particular original equation check，不以小數近似替代。",
          "en": "Recompute Dioph particular original equation check with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1260",
          "1890"
        ],
        "output": "630",
        "label": "Dioph particular original equation check"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Dioph normalized original a over gcd := div(-90, 45)",
      "after": "Dioph normalized original a over gcd = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph normalized original a over gcd，不以小數近似替代。",
          "en": "Recompute Dioph normalized original a over gcd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-90",
          "45"
        ],
        "output": "-2",
        "label": "Dioph normalized original a over gcd"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Dioph normalized original b over gcd := div(135, 45)",
      "after": "Dioph normalized original b over gcd = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph normalized original b over gcd，不以小數近似替代。",
          "en": "Recompute Dioph normalized original b over gcd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "135",
          "45"
        ],
        "output": "3",
        "label": "Dioph normalized original b over gcd"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Dioph signed general y step := mul(-1, -2)",
      "after": "Dioph signed general y step = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph signed general y step，不以小數近似替代。",
          "en": "Recompute Dioph signed general y step with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-2"
        ],
        "output": "2",
        "label": "Dioph signed general y step"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Dioph homogeneous a times x step := mul(-90, 3)",
      "after": "Dioph homogeneous a times x step = -270",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph homogeneous a times x step，不以小數近似替代。",
          "en": "Recompute Dioph homogeneous a times x step with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-90",
          "3"
        ],
        "output": "-270",
        "label": "Dioph homogeneous a times x step"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Dioph homogeneous b times y step := mul(135, 2)",
      "after": "Dioph homogeneous b times y step = 270",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph homogeneous b times y step，不以小數近似替代。",
          "en": "Recompute Dioph homogeneous b times y step with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "135",
          "2"
        ],
        "output": "270",
        "label": "Dioph homogeneous b times y step"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Dioph full homogeneous step cancellation := sum(-270, 270)",
      "after": "Dioph full homogeneous step cancellation = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留正負號、分母、索引及所在載體。",
        "en": "Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Dioph full homogeneous step cancellation，不以小數近似替代。",
          "en": "Recompute Dioph full homogeneous step cancellation with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-270",
          "270"
        ],
        "output": "0",
        "label": "Dioph full homogeneous step cancellation"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "particular original pair=[14, 14] solves c=630;normalized coprime A=-2,B=3;step pair=[3, 2]",
      "after": "for EVERY integer t,x=x0+(3)t,y=y0+(2)t gives left side=c+t*0=c;all displayed solutions valid",
      "relation": "implies",
      "rule_id": "dioph_every_integer_parameter_sufficiency",
      "conditions": [],
      "explanation": {
        "zh-Hant": "參數t可任意正負整數，原方程逐代入常數項c與係數0，所以全部參數值有效；這不是只列幾個t樣本。",
        "en": "The parameter t is any signed integer.Substitution leaves the constant term c and coefficient0 of t,so every parameter value works,not merely a few samples."
      },
      "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": "ANY other integer solution;differences dx=x-x0,dy=y-y0 satisfy A*dx+B*dy=0;A=-2,B=3;source Bezout gives1*A+1*B=1",
      "after": "dx=B*(-(1)*dy+(1)*dx),so t=dx/B is an INTEGER;B!=0 then dy=-A*t;ALL solutions have the displayed form",
      "relation": "implies",
      "rule_id": "dioph_all_integer_solutions_necessity_coprime_not_grid",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把正規化Bézout乘dx，再用A·dx=−B·dy，直接得到B整除dx。B非零才作除法，並從方程消去B得dy=−A·t，故沒有漏掉任何整數解。",
        "en": "Multiply the normalized Bezout identity by dx and substitute A*dx=−B*dy to show B divides dx.Divide only when B is nonzero,then cancel B in the equation to obtain dy=−A*t.This accounts for every integer solution."
      },
      "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": [
      14,
      14,
      3,
      2,
      "x=14+(3)t",
      "y=14+(2)t",
      "all_and_only_integer_solutions_for_t_in_Z"
    ]
  },
  "conclusion": {
    "canonical_value": "(14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations/questions.jsonl",
      "batch_sha256": "8f24af0604a2e706355a3ac6ed118be11f0a925360c15eeb4935e5d76c1f7370",
      "original_line": 64,
      "original_line_sha256": "f8e685ff9e94c762723b2ad7cbe9c2d7b463c4fdf93ecdc576c4f67540b30809",
      "question_compact_sha256": "cbca5033bd88cf4483e307e6762415892009d6fead99a75cba8b6bccf0a1da0e",
      "english_row_compact_sha256": "b5abef82364319734276144945d0d5dbc98739b8146ffbc44dd56fabb8066cfe",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-diophantine-equation": {
        "sha256": "f0c63a1636cb208e054adfa59fdc5cb29078971741ce31fd2c59710199df6a79",
        "english_sha256": "77cad5a116f985108c11f12cd56c1fc7c6f88ab25c327092ad4ad569e0f44a3d",
        "original_path": "public/data/mko/linear-diophantine-equation.json",
        "english_path": "public/data/i18n/editions/mko-linear-diophantine-equation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9ed24319f1e4df0dd58d2a2dfbf2846def03f020cb5ebf70613fab53c9165f11",
    "original_candidate_sha256": "4faf1cb18da6fb53b4fc2a3935a4625b61a0448f561915e0f99c6f595a27718f",
    "original_candidate_line_sha256": "4faf1cb18da6fb53b4fc2a3935a4625b61a0448f561915e0f99c6f595a27718f"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p39",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612",
      "question_first_witness_sha256": "c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 47,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "6a0e9b4b9549db4ebe94bd4260df7fecf5a53bf316da7e92529a168804c9fb98",
      "arithmetic_steps": 38
    },
    "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."
  }
}