{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261106-r06-linear-diophantine-equation-0095",
  "pilot_number": "R0038-19620",
  "number": 19620,
  "family_id": "qf-mod37-dioph-degenerate-boundary",
  "title": {
    "zh-Hant": "第 19620 題：線性Diophantine方程",
    "en": "Question 19620: Double-Zero Coefficients Must Not Divide by Gcd Zero"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261106-r06-linear-diophantine-equation-0095",
    "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-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(7, 否, no_integer_solution, never_divide_by_gcd_zero)",
      "explanation_zh": "Wave5封閉層核對=(7, 否, no_integer_solution, never_divide_by_gcd_zero)；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": 3,
      "skills": [
        "線性Diophantine方程"
      ]
    },
    "variant": {
      "family_id": "qf-mod37-dioph-degenerate-boundary",
      "seed": 202611066095,
      "parameters": {
        "a": 0,
        "b": 0,
        "c": 7,
        "domain": "Exact_integer_linear_Diophantine_equation",
        "operation": "dioph-degenerate-boundary",
        "semantic_template": "qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":7}"
      }
    },
    "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-0095",
    "batch_id": "batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "33a9bfd46aaa679b1c2802b1fc212d7c987e33faa2a7eb33184fb24f7e2bc580",
      "family_id": "qf-mod37-dioph-degenerate-boundary",
      "seed": 202611066095,
      "semantic_template": "qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":7}"
    },
    "fields": {
      "title": "Double-Zero Coefficients Must Not Divide by Gcd Zero",
      "stem": "Use exact original integers a=0, b=0, c=7 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 (c, solvable, solution_set, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(7, No, no_integer_solution, never_divide_by_gcd_zero)",
        "explanation": "Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (7, No, no_integer_solution, never_divide_by_gcd_zero)."
      },
      "skills": [
        "Double-Zero Coefficients Must Not Divide by Gcd Zero"
      ],
      "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-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use exact original integers a=0, b=0, c=7 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 (c, solvable, solution_set, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=dioph-degenerate-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[c, solvable, solution_set, boundary]",
      "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": "ACTUAL original equation0*x+0*y=7;BOTH coefficients0",
      "after": "left side is0 for EVERY integer pair;solvable iff c=0=否;solution set=no_integer_solution;NO gcd0 division",
      "relation": "implies",
      "rule_id": "dioph_double_zero_all_pairs_or_none_not_one_parameter_line",
      "conditions": [],
      "explanation": {
        "zh-Hant": "雙零時方程左邊恆0；c=0是全部Z²，不是一條t參數線，c非0則全無解。不要計c/gcd或對gcd0取mod。",
        "en": "With both coefficients zero the left side is identically0.For c=0 all of Z² solves it,not one parameter line;otherwise none does.Never divide by or reduce modulo gcd0."
      },
      "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": [
      7,
      false,
      "no_integer_solution",
      "never_divide_by_gcd_zero"
    ]
  },
  "conclusion": {
    "canonical_value": "(7, 否, no_integer_solution, never_divide_by_gcd_zero)",
    "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": {
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    "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": 95,
      "original_line_sha256": "c10af9504bbdc474b6f40815d1a88fcd27c8b040bc24fb62e38ef9762bc1db01",
      "question_compact_sha256": "33a9bfd46aaa679b1c2802b1fc212d7c987e33faa2a7eb33184fb24f7e2bc580",
      "english_row_compact_sha256": "2715a5444786919c0c45c8d53462682e1ee738821aa074d5b23503d0e4a16cb4",
      "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"
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        "original_path": "public/data/mko/linear-diophantine-equation.json",
        "english_path": "public/data/i18n/editions/mko-linear-diophantine-equation.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9ed24319f1e4df0dd58d2a2dfbf2846def03f020cb5ebf70613fab53c9165f11",
    "original_candidate_sha256": "64826e6a3531eb55597915f2f13a314c31c1494e071483303311c18e7ce31a9c",
    "original_candidate_line_sha256": "64826e6a3531eb55597915f2f13a314c31c1494e071483303311c18e7ce31a9c"
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  "historical_source_state": {
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      "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": {
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    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 2,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "1d0d160185ba296995639c23a29092cbb25e33407026640d17b3a7cad0323c2d",
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    "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."
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}