{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r05-modular-congruence-0001",
  "pilot_number": "R0017-09076",
  "number": 9076,
  "family_id": "qf-ext18-modeq-remainder-gate",
  "title": {
    "zh-Hant": "第 9076 題：正模數同餘",
    "en": "Question 9076: Positive-Modulus Guard before Natural Remainder Comparison"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r05-modular-congruence-0001",
    "batch_id": "batch-20261013-advance-20261018-r05-modular-congruence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-modular-congruence"
    ],
    "title_zh": "正模數同餘",
    "stem_zh": "固定模數。操作=modeq-remainder-gate；資料={\"n\":0,\"a\":7,\"b\":12}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 7, 12, 否, 不求值, 不求值)",
      "explanation_zh": "擴展基礎核對=(0, 7, 12, 否, 不求值, 不求值)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "正模數同餘"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-modeq-remainder-gate",
      "seed": 202610185001,
      "parameters": {
        "n": 0,
        "a": 7,
        "b": 12,
        "domain": "Nat_positive_mod_congruence",
        "operation": "modeq-remainder-gate",
        "semantic_template": "qf-ext18-modeq-remainder-gate/{\"a\":7,\"b\":12,\"n\":0}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261018-r05-modular-congruence-0001",
    "batch_id": "batch-20261013-advance-20261018-r05-modular-congruence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4d5aae40cc8ccab5a2f4de483e1e4fb75abfda0f204687e1ee3ad4f6d6c3138e",
      "family_id": "qf-ext18-modeq-remainder-gate",
      "seed": 202610185001,
      "semantic_template": "qf-ext18-modeq-remainder-gate/{\"a\":7,\"b\":12,\"n\":0}"
    },
    "fields": {
      "title": "Positive-Modulus Guard before Natural Remainder Comparison",
      "stem": "Fix modulus n=0, natural a=7, b=12. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.",
      "answer": {
        "kind": "expression",
        "value": "(0, 7, 12, No, not-evaluated, not-evaluated)",
        "explanation": "Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 7, 12, No, not-evaluated, not-evaluated)."
      },
      "skills": [
        "Positive-Modulus Guard before Natural Remainder Comparison"
      ],
      "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": "固定模數。操作=modeq-remainder-gate；資料={\"n\":0,\"a\":7,\"b\":12}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。",
        "en": "Fix modulus n=0, natural a=7, b=12. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "fixed natural modulus candidate=0",
      "after": "positive modulus=否",
      "relation": "implies",
      "rule_id": "fixed_positive_modulus_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同餘在同一正模數下以餘數比較，不等於數值相等。乘法保持同餘不表示可無條件取消共同因子。",
        "en": "Congruence compares remainders at one positive modulus,not numerical equality.Multiplicative preservation does not permit unconditional cancellation of a common factor."
      },
      "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": "n=0;a=7;b=12",
      "after": "remainders=不求值;congruence=不求值",
      "relation": "implies",
      "rule_id": "zero_modulus_no_remainder_evaluation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "模數不為正時兩個餘數與同餘欄均不求值；不是由API預設餘數製造真假。",
        "en": "When the modulus is not positive,both remainders and congruence remain unevaluated.Do not manufacture a truth value from an API default."
      },
      "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": [
      0,
      7,
      12,
      false,
      "不求值",
      "不求值"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 7, 12, 否, 不求值, 不求值)",
    "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-20261013-advance-20261018-r05-modular-congruence-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261018-r05-modular-congruence-foundations/questions.jsonl",
      "batch_sha256": "11ab2f76ce5fc3b4ea3ee6beb1fac6bbbec928d3634a3cf0d7d08dbbe78db498",
      "original_line": 1,
      "original_line_sha256": "572f10ce8b8445ffaace286c4974f759b1c8b6d383fea79d4ddd6511ace245f3",
      "question_compact_sha256": "4d5aae40cc8ccab5a2f4de483e1e4fb75abfda0f204687e1ee3ad4f6d6c3138e",
      "english_row_compact_sha256": "0c5dbc4efdf8b8b901621af7c09453d5093e9e0d27b702e643ada39546fbf1ec",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261018-r05-modular-congruence-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261018-r05-modular-congruence-foundations.en.jsonl"
    },
    "mko_bindings": {
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        "english_sha256": "b1c4c24d7715216387b6c6d80d0393707e999f09f844c40b624b56ec79f8ec35",
        "original_path": "public/data/mko/modular-congruence.json",
        "english_path": "public/data/i18n/editions/mko-modular-congruence.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "20e59f30c0b6218e6d9614c16170d770e56a2f1e4ae5ce39d0a8a3db206afa61",
    "original_candidate_sha256": "6bac9c29b98345915842d41e6818cc785c5245a1c2c3d262d8a0b1083b7f1927",
    "original_candidate_line_sha256": "6bac9c29b98345915842d41e6818cc785c5245a1c2c3d262d8a0b1083b7f1927"
  },
  "historical_source_state": {
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      "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-20261026-006-p18",
    "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": "abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8",
      "question_first_witness_sha256": "43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04"
    },
    "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": "39b84f01926195cae572a3fb785bb463aa36f68ae04b206607edda54f9fd797a",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}