{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260918-finite-event-cardinality-0025",
  "pilot_number": "R0062-31925",
  "number": 31925,
  "family_id": "qf-finite-event-cardinality",
  "title": {
    "zh-Hant": "第 31925 題：qf-finite-event-cardinality",
    "en": "Question 31925: Cardinality of a Finite Event"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260918-finite-event-cardinality-0025",
    "batch_id": "batch-20260918-finite-events",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-event"
    ],
    "title_zh": "qf-finite-event-cardinality",
    "stem_zh": "S={97, 98, 99}，F=P(S)，E={98}。E 的基數是多少？",
    "answer": {
      "kind": "numeric",
      "value": 1,
      "explanation_zh": "E 有 1 個不同結果；這不是機率。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "事件基數"
      ]
    },
    "variant": {
      "family_id": "qf-finite-event-cardinality",
      "seed": 20261243,
      "parameters": {
        "S": [
          97,
          98,
          99
        ],
        "E": [
          98
        ],
        "F": "P(S)",
        "semantic_template": "qf-finite-event-cardinality/{\"E\":[98],\"F\":\"P(S)\",\"S\":[97,98,99]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "schema_pending",
        "independent_recomputation_pending"
      ],
      "warning_zh": "本題是有限候選重算；不構成普遍證明、形式 Evidence 或人工審查。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-09-18",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260918-finite-event-cardinality-0025",
    "batch_id": "batch-20260918-finite-events",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "cec267b81e96915976eefefc54fe7afc8180569362441b0312fe898dbe2212d2",
      "family_id": "qf-finite-event-cardinality",
      "seed": 20261243,
      "semantic_template": "qf-finite-event-cardinality/{\"E\":[98],\"F\":\"P(S)\",\"S\":[97,98,99]}"
    },
    "fields": {
      "title": "Cardinality of a Finite Event",
      "stem": "Let S={97, 98, 99}, F=P(S), and E={98}. What is the cardinality of E?",
      "answer": {
        "kind": "numeric",
        "value": 1,
        "explanation": "E contains 1 distinct outcome. This is a cardinality, not a probability."
      },
      "skills": [
        "event cardinality"
      ],
      "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": "S={97, 98, 99}，F=P(S)，E={98}。E 的基數是多少？",
        "en": "Let S={97, 98, 99}, F=P(S), and E={98}. What is the cardinality of E?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-finite-event-cardinality, no-domain-field, no-operation-field, no-form-field]",
      "after": "derived = [[cardinality_NUMBER_not_probability]]",
      "relation": "implies",
      "rule_id": "R62_EXACT_source_keys_domain_operation_form_AND_output_types",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原bank核欄位：BASE方程用form及論域，BASE事件／補集無operation，階乘無domain；ext18用local-hyphen，其他legacy用underscore。輸出型別不統一成tuple或fraction。",
        "en": "Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [S original sample, [97, 98, 99]]",
      "after": "derived = [[97, 98, 99]]",
      "relation": "implies",
      "rule_id": "BASE_FINITE_set_distinct_INTEGER_list_no_silent_duplicate_fix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原BASE有限集合是distinct整數list，重複輸入不偷偷去重修成合法。數值排序只用於衍生顯示，原資料不變；不搬來expectation的100元素或SAFE界。",
        "en": "The BASE finite set is a DISTINCT integer list;do not silently repair duplicates.Sort numerically ONLY for derived display,retaining original data.Do not import the expectation100-element or SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [E original event candidate, [98]]",
      "after": "derived = [[98]]",
      "relation": "implies",
      "rule_id": "BASE_FINITE_set_distinct_INTEGER_list_no_silent_duplicate_fix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原BASE有限集合是distinct整數list，重複輸入不偷偷去重修成合法。數值排序只用於衍生顯示，原資料不變；不搬來expectation的100元素或SAFE界。",
        "en": "The BASE finite set is a DISTINCT integer list;do not silently repair duplicates.Sort numerically ONLY for derived display,retaining original data.Do not import the expectation100-element or SAFE bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [0, 98, [97, 98, 99]]",
      "after": "derived = [是]",
      "relation": "implies",
      "rule_id": "Event_EACH_candidate_element_membership_in_ORIGINAL_S",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐候選事件元素核對是否在原S，空事件的所有元素在S條件真。F=P(S)與概率law不同，不能由powerset字樣推等可能。",
        "en": "Check EACH candidate event element in ORIGINAL S.The empty event satisfies the subset condition vacuously.F=P(S)is not a probability law and does not imply equal likelihood."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [[97, 98, 99], [98], P(S)]",
      "after": "derived = [[是], [], 是]",
      "relation": "implies",
      "rule_id": "Event_FULLpowerset_qualification_ALL_subset_flags_NOT_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "F明示P(S)，事件資格恰為E⊆S；subset-validity的超域E是合法No案例，不可先刪掉反例。元素數量、事件發生與概率是不同問題。",
        "en": "With EXACT F=P(S),event qualification is precisely E⊆S.An outside E is a LEGAL No in subset-validity;do not delete the witness first.Cardinality,occurrence and probability are distinct questions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Event cardinality ORIGINAL distinct element slot0 := sum(0, 1)",
      "after": "Event cardinality ORIGINAL distinct element slot0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Event cardinality ORIGINAL distinct element slot0，不以小數近似替代。",
          "en": "Recompute Event cardinality ORIGINAL distinct element slot0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Event cardinality ORIGINAL distinct element slot0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [[97, 98, 99], [98]]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "Event_CARDINALITY_NUMBER_distinct_count_NO_probability_or_nonempty_S",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐個distinct元素數出|E|，答案是NUMBER不是fraction或概率。空事件基數0，即使S也空仍可；不假設uniform分布。",
        "en": "Count EACH distinct element for|E|.The answer is a NUMBER,not a fraction or probability.The empty event has cardinality0,even when S is empty.No uniform distribution is assumed."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": 1
  },
  "conclusion": {
    "canonical_value": "1",
    "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-20260918-finite-events",
      "batch_path": "public/data/questions/batches/batch-20260918-finite-events/questions.jsonl",
      "batch_sha256": "5afdb8e28e1d24a99739443746e81c96f3b1342395fffe3676e3a0844c6bae1d",
      "original_line": 100,
      "original_line_sha256": "b38095251003ca45dbcfbc85d1c6c1ac341154bb6649e86035a7925e12145d27",
      "question_compact_sha256": "cec267b81e96915976eefefc54fe7afc8180569362441b0312fe898dbe2212d2",
      "english_row_compact_sha256": "d442746ca0ee94f9559a3f9b4af483b5dfd4dcd1cffa0d86b97a051879a3ac3e",
      "english_manifest_path": "public/data/i18n/editions/batch-20260918-finite-events.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20260918-finite-events.en.jsonl"
    },
    "mko_bindings": {
      "mko-event": {
        "sha256": "5962ae6672cb79c34888bdda4ac66de557a4f9b21243f2445414d7dd326b9c74",
        "english_sha256": "12d8199c90226e08b52a3f9cf27ce44053dc8473e8bab4280145aea146a4c261",
        "original_path": "public/data/mko/event.json",
        "english_path": "public/data/i18n/editions/mko-event.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "59d384ca3d958b27165f4295ff96efb155e7b12d3fb70bcdd0ed68148513c05a",
    "original_candidate_sha256": "f4c8bbafa9abfcc065bd7dceaf9df102ae7adcd9716968ecc05c088e8f11da9b",
    "original_candidate_line_sha256": "f4c8bbafa9abfcc065bd7dceaf9df102ae7adcd9716968ecc05c088e8f11da9b"
  },
  "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-20261026-011-p63",
    "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": "963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd",
      "question_first_witness_sha256": "d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "a627cdef47d90beb8922c69653ebe40592d9af103fb4e925c2dee8bc0615f83e",
      "arithmetic_steps": 1
    },
    "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."
  }
}