{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r07-quantifier-0081",
  "pilot_number": "R0007-04356",
  "number": 4356,
  "family_id": "qf-density10-quantifier-nested-counts",
  "title": {
    "zh-Hant": "第 4356 題：兩種有序嵌套枚舉的計數",
    "en": "Question 4356: Two Ordered Nested-Quantifier Counts"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r07-quantifier-0081",
    "batch_id": "batch-20261008-advance-20261010-r07-quantifier-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-quantifier"
    ],
    "title_zh": "兩種有序嵌套枚舉的計數",
    "stem_zh": "有限明示論域 U=[-1]；V=[100]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。",
    "answer": {
      "kind": "expression",
      "value": "(0, 0)",
      "explanation_zh": "有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "兩種有序嵌套枚舉的計數"
      ]
    },
    "variant": {
      "family_id": "qf-density10-quantifier-nested-counts",
      "seed": 202610107081,
      "parameters": {
        "U": [
          -1
        ],
        "V": [
          100
        ],
        "relation": "eq",
        "operation": "density10_quantifier_nested_counts",
        "domain": "finite_integer_domain",
        "semantic_template": "qf-density10-quantifier-nested-counts/[1,1,0]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261010-r07-quantifier-0081",
    "batch_id": "batch-20261008-advance-20261010-r07-quantifier-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "aaf844a987fe9511783f8b065235a8760eed4d11c983cc589b7b102d05cdb334",
      "family_id": "qf-density10-quantifier-nested-counts",
      "seed": 202610107081,
      "semantic_template": "qf-density10-quantifier-nested-counts/[1,1,0]"
    },
    "fields": {
      "title": "Two Ordered Nested-Quantifier Counts",
      "stem": "U=[-1], V=[100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).",
      "answer": {
        "kind": "expression",
        "value": "(0, 0)",
        "explanation": "Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."
      },
      "skills": [
        "Two Ordered Nested-Quantifier Counts"
      ],
      "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": "有限明示論域 U=[-1]；V=[100]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。",
        "en": "U=[-1], V=[100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "complete finite U=[-1]",
      "after": "universal on empty domain=True;existential on empty domain=False",
      "relation": "implies",
      "rule_id": "finite_quantifier_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只枚舉明示有限論域，空域全稱真、存在假；完成有限枚舉不外推無限論域。",
        "en": "Enumerate only the declared finite domain;empty-domain universals are true and existentials false.Complete finite enumeration does not extend to an infinite domain."
      },
      "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": "x=-1;y=100;relation=eq",
      "after": "否",
      "relation": "implies",
      "rule_id": "finite_relation_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整代入此有序對，不把x與y交換。",
        "en": "Evaluate this ordered pair without exchanging x and y."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "x=-1;row truths=[否]",
      "after": "some y satisfies=否",
      "relation": "implies",
      "rule_id": "nested_exists_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "內層存在需至少一個真項；V為空時每個x的此條件皆假。",
        "en": "The inner existential needs at least one true entry;when V is empty this condition is false for every x."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "y=100;column truths=[否]",
      "after": "all x satisfy=否",
      "relation": "implies",
      "rule_id": "nested_forall_column",
      "conditions": [],
      "explanation": {
        "zh-Hant": "內層全稱需全部為真；U為空時每個y的此條件皆真。",
        "en": "The inner universal requires every entry true;when U is empty it holds for each y."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "exists-by-x=[否];forall-by-y=[否]",
      "after": "(0, 0)",
      "relation": "implies",
      "rule_id": "nested_quantifier_count_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一欄計有某個y可配的x數，第二欄計對所有x都符合的y數；兩者不是交換量詞後同一件事。",
        "en": "The first count concerns x with some suitable y;the second concerns y satisfying every x.These are distinct ordered quantified conditions."
      },
      "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,
      0
    ],
    "relation_matrix": [
      [
        false
      ]
    ],
    "exists_by_x": [
      false
    ],
    "forall_by_y": [
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 0)",
    "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-20261008-advance-20261010-r07-quantifier-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r07-quantifier-foundations/questions.jsonl",
      "batch_sha256": "13b0fa8d8039c9cb766931f8811a087808fd7708cee37eff0ae37506a6f390f1",
      "original_line": 81,
      "original_line_sha256": "e5e05d3f3b293582c850eb88cb8c3e103cac46dfc6f404701bd3a7067730255f",
      "question_compact_sha256": "aaf844a987fe9511783f8b065235a8760eed4d11c983cc589b7b102d05cdb334",
      "english_row_compact_sha256": "8cf08a98230b16d24f0842f7bd096faf658abfb6e500b2544f4623275826acac",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r07-quantifier-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r07-quantifier-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-quantifier": {
        "sha256": "3b2f73bd57aa43b5441edef2b27d44470f8cbf04a229fb31e98243708e818341",
        "english_sha256": "cf34ab77acaa732f1cc6926b00bee27e73188f1108aaad401ca0a12fc391cc57",
        "original_path": "public/data/mko/quantifier.json",
        "english_path": "public/data/i18n/editions/mko-quantifier.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "8333da6c00fab92796a8841f1c1ed08093697505af543e61426e5aa20e9b8f9a",
    "original_candidate_line_sha256": "8333da6c00fab92796a8841f1c1ed08093697505af543e61426e5aa20e9b8f9a"
  },
  "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-20261026-004-p8",
    "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": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a",
      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "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": "afaf3ae351adf925f7b138be50c0c825221be2a0dac3b840e3d27bb75c4d55f6",
      "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."
  }
}