{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r05-modular-congruence-0026",
  "pilot_number": "R0017-09026",
  "number": 9026,
  "family_id": "qf-ext18-modeq-equivalence-class",
  "title": {
    "zh-Hant": "第 9026 題：正模數同餘",
    "en": "Question 9026: A Natural Congruence Class in an Inclusive Window"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r05-modular-congruence-0026",
    "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-equivalence-class；資料={\"n\":2,\"a\":1,\"bound\":25}。請依 [n,a,bound,remainder,class_members] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])",
      "explanation_zh": "擴展基礎核對=(2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])；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-equivalence-class",
      "seed": 202610185026,
      "parameters": {
        "n": 2,
        "a": 1,
        "bound": 25,
        "domain": "Nat_positive_mod_congruence",
        "operation": "modeq-equivalence-class",
        "semantic_template": "qf-ext18-modeq-equivalence-class/{\"a\":1,\"bound\":25,\"n\":2}"
      }
    },
    "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-0026",
    "batch_id": "batch-20261013-advance-20261018-r05-modular-congruence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "118b375df7069f38acd77af452b01ac87232cecafca8edf15946286b8be86363",
      "family_id": "qf-ext18-modeq-equivalence-class",
      "seed": 202610185026,
      "semantic_template": "qf-ext18-modeq-equivalence-class/{\"a\":1,\"bound\":25,\"n\":2}"
    },
    "fields": {
      "title": "A Natural Congruence Class in an Inclusive Window",
      "stem": "Fix positive modulus n=2, natural a=1, bound=25. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).",
      "answer": {
        "kind": "expression",
        "value": "(2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])",
        "explanation": "Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])."
      },
      "skills": [
        "A Natural Congruence Class in an Inclusive Window"
      ],
      "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-equivalence-class；資料={\"n\":2,\"a\":1,\"bound\":25}。請依 [n,a,bound,remainder,class_members] 的欄序回答。",
        "en": "Fix positive modulus n=2, natural a=1, bound=25. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "fixed natural modulus candidate=2",
      "after": "positive modulus=是",
      "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": "representative remainder:natural dividend=1;positive divisor=2",
      "after": "1=0·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "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": {
        "operation": "euclidean_division",
        "inputs": [
          "1",
          "2",
          "0"
        ],
        "output": "1",
        "label": "representative remainder"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "window bound minus residue := sub(25, 1)",
      "after": "window bound minus residue = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步原始操作數及指定次序精確計算，不用近似值或替換候選。",
        "en": "Calculate exactly in the specified source-operand order without approximation or substituting a different candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 window bound minus residue，不以小數近似替代。",
          "en": "Recompute window bound minus residue with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "25",
          "1"
        ],
        "output": "24",
        "label": "window bound minus residue"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "last progression index:natural dividend=24;positive divisor=2",
      "after": "24=12·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "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": {
        "operation": "euclidean_division",
        "inputs": [
          "24",
          "2",
          "12"
        ],
        "output": "0",
        "label": "last progression index"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "residue=1;modulus=2;0<=k<=12",
      "after": "all class members=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]",
      "relation": "implies",
      "rule_id": "complete_residue_progression",
      "conditions": [],
      "explanation": {
        "zh-Hant": "Euclid唯一性給x≡a當且僅當x=r+k*n；自然x且0≤r<n強迫k≥0，上界給k≤floor((bound−r)/n)。完整保留兩端，不只找幾個樣本。",
        "en": "Euclidean uniqueness gives x≡a exactly when x=r+k*n.Natural x and0≤r<n force k≥0;the bound gives k≤floor((bound−r)/n).Retain the full inclusive progression,not a sample."
      },
      "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": [
      2,
      1,
      25,
      1,
      [
        1,
        3,
        5,
        7,
        9,
        11,
        13,
        15,
        17,
        19,
        21,
        23,
        25
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])",
    "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": 26,
      "original_line_sha256": "4049582fde15d0966ae2dd0b769c351e63f5d4eb875a14ca8f7c239608f47dad",
      "question_compact_sha256": "118b375df7069f38acd77af452b01ac87232cecafca8edf15946286b8be86363",
      "english_row_compact_sha256": "86869875e09de09c27e0caa3cbe05254ff15b2a34b305476cba7db13c8eb05d5",
      "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": {
      "mko-modular-congruence": {
        "sha256": "f2701dae3fc0037b15acf2146bea327e803c9f727d640db0a85d02de29f3f635",
        "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": "4bcc825be0686f5f1d406b042de4f6639e64034107b7f34e29bd7d0bf18cf8ae",
    "original_candidate_line_sha256": "4bcc825be0686f5f1d406b042de4f6639e64034107b7f34e29bd7d0bf18cf8ae"
  },
  "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-20261026-006-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": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "652862b23d289bc9f23537db7b7fbb5329a873ab696c400b5bda4aa830e80dba",
      "arithmetic_steps": 3
    },
    "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."
  }
}