{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261031-r01-at-most-countable-set-0069",
  "pilot_number": "R0030-15869",
  "number": 15869,
  "family_id": "qf-mod31-countable-enumeration",
  "title": {
    "zh-Hant": "第 15869 題：至多可數集",
    "en": "Question 15869: Keep Enumeration Direction and Original List Order Explicit"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261031-r01-at-most-countable-set-0069",
    "batch_id": "batch-20261014-advance-20261031-r01-at-most-countable-set-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-at-most-countable-set"
    ],
    "title_zh": "至多可數集",
    "stem_zh": "有限完整清單或明示全稱注入證書。操作=countable-enumeration；資料={\"elements\":[18,68,-19],\"index\":0}。請依 [index,element,enumeration_pairs,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete)",
      "explanation_zh": "Wave4收束核對=(0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete)；依完整明示資料重算；有限與可數無限不可混同；單一失敗encoding不證不可數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "至多可數集"
      ]
    },
    "variant": {
      "family_id": "qf-mod31-countable-enumeration",
      "seed": 202610311069,
      "parameters": {
        "elements": [
          18,
          68,
          -19
        ],
        "index": 0,
        "domain": "Explicit_at_most_countable_certificate",
        "operation": "countable-enumeration",
        "semantic_template": "qf-mod31-countable-enumeration/{\"elements\":[-19,18,68],\"index\":0}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r01-at-most-countable-set-0069",
    "batch_id": "batch-20261014-advance-20261031-r01-at-most-countable-set-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "dc8a453b349bd3e6b4bc6328e3744dcd6a0e4b424ffa962dbe881eb3604968c6",
      "family_id": "qf-mod31-countable-enumeration",
      "seed": 202610311069,
      "semantic_template": "qf-mod31-countable-enumeration/{\"elements\":[-19,18,68],\"index\":0}"
    },
    "fields": {
      "title": "Keep Enumeration Direction and Original List Order Explicit",
      "stem": "Complete nonempty duplicate-free finite integer enumeration elements=[18, 68, -19], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.",
      "answer": {
        "kind": "expression",
        "value": "(0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete)",
        "explanation": "Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete)."
      },
      "skills": [
        "Keep Enumeration Direction and Original List Order Explicit"
      ],
      "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": "有限完整清單或明示全稱注入證書。操作=countable-enumeration；資料={\"elements\":[18,68,-19],\"index\":0}。請依 [index,element,enumeration_pairs,certificate] 欄序回答。",
        "en": "Complete nonempty duplicate-free finite integer enumeration elements=[18, 68, -19], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "DECLARED complete finite integer set in original list order=[18, 68, -19];length=3",
      "after": "nonempty duplicate-free registered family;original order retained;empty-set countability not denied",
      "relation": "implies",
      "rule_id": "countable_complete_finite_list_premise_not_sampled_subset",
      "conditions": [],
      "explanation": {
        "zh-Hant": "清單的完整性是題目明示集合前提，不從有限樣本推未知全集；不排序、不刪重複。此題族拒絕空表只是介面限制，空集仍至多可數。",
        "en": "Completeness is the declared set premise,not inferred for an unknown universe from samples.Keep order and reject duplicates rather than sorting or deduplicating.The family's empty-list rejection is an interface limit,not denial of empty-set countability."
      },
      "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": "complete original ordered enumeration=[18, 68, -19];selected zero-based index=0",
      "after": "selected element=18;position_to_element pairs=[[0, 18], [1, 68], [2, -19]]",
      "relation": "implies",
      "rule_id": "finite_countable_enumeration_order_not_injection_reversal",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原位置讀元素，列完整位置到元素的方向；這個有限enumeration證明有限而至多可數，不把有限個元素稱可數無限。",
        "en": "Read the original position and list every position-to-element pair.This finite enumeration supplies a finite and thus at-most-countable certificate,not countable infinitude."
      },
      "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,
      18,
      [
        [
          0,
          18
        ],
        [
          1,
          68
        ],
        [
          2,
          -19
        ]
      ],
      "finite_enumeration_complete"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete)",
    "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-20261031-r01-at-most-countable-set-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations/questions.jsonl",
      "batch_sha256": "7b382f15a68b0f6e0c1e44dc793ede056d8bd39aff2ca4e878a95654740ee01a",
      "original_line": 69,
      "original_line_sha256": "14143b6925a55f6c825d4cae90d86bef1323ff09bea512f4b9b79f2fc785b05d",
      "question_compact_sha256": "dc8a453b349bd3e6b4bc6328e3744dcd6a0e4b424ffa962dbe881eb3604968c6",
      "english_row_compact_sha256": "bc7e5efc7f61241d618c6eb39a7f3f9012161e509b6412180823cd41b7a608d6",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-at-most-countable-set": {
        "sha256": "9c7748686b7974341920a59d07628f20017ef6d65476d751b2feb7424f7b3b57",
        "english_sha256": "a73faf7f1c04553efad774692eab18a2de59165f0fedaac3ba03ee75c7276def",
        "original_path": "public/data/mko/at-most-countable-set.json",
        "english_path": "public/data/i18n/editions/mko-at-most-countable-set.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d85118d74859ec7e59736714f767732c6ec092eaa0bf5279abeb28e4a628e3c3",
    "original_candidate_sha256": "775b7e3025a3ecba40d1f4b68bb72c982204f34e01ba4f1d8cd92f64dd64c9b9",
    "original_candidate_line_sha256": "775b7e3025a3ecba40d1f4b68bb72c982204f34e01ba4f1d8cd92f64dd64c9b9"
  },
  "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-20261026-007-p31",
    "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": "73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc",
      "question_first_witness_sha256": "ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336"
    },
    "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": "2bf6100c40ae6f875e123fb34666666ee8b8d2b96c78cbef877ec5fb143d2d46",
      "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."
  }
}