{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261012-r03-proposition-0026",
  "pilot_number": "R0010-05751",
  "number": 5751,
  "family_id": "qf-density12-proposition-open-binding",
  "title": {
    "zh-Hant": "第 5751 題：自由變數語句的封閉資格",
    "en": "Question 5751: Open-Sentence Binding and Truth Eligibility"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261012-r03-proposition-0026",
    "batch_id": "batch-20261008-advance-20261012-r03-proposition-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-proposition"
    ],
    "title_zh": "自由變數語句的封閉資格",
    "stem_zh": "古典整數語境；語句x≤-2，assignment=null（未綁定）。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。",
    "answer": {
      "kind": "expression",
      "value": "(否, 未封閉, 不適用)",
      "explanation_zh": "封閉資格與代入真假=(否, 未封閉, 不適用)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "自由變數語句的封閉資格"
      ]
    },
    "variant": {
      "family_id": "qf-density12-proposition-open-binding",
      "seed": 202610123026,
      "parameters": {
        "threshold": -2,
        "assignment": null,
        "operation": "density12_proposition_open_binding",
        "domain": "integer_open_sentence_binding",
        "semantic_template": "qf-density12-proposition-open-binding/[\"open_sentence\"]"
      }
    },
    "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-20261012-r03-proposition-0026",
    "batch_id": "batch-20261008-advance-20261012-r03-proposition-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0213658fad3206e7b0ee2063567e34f2c15c4d2cf79a36b1187d7704aa0d11e5",
      "family_id": "qf-density12-proposition-open-binding",
      "seed": 202610123026,
      "semantic_template": "qf-density12-proposition-open-binding/[\"open_sentence\"]"
    },
    "fields": {
      "title": "Open-Sentence Binding and Truth Eligibility",
      "stem": "In the classical integer context, consider x≤-2 with assignment=null (unbound). Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).",
      "answer": {
        "kind": "expression",
        "value": "(No, open, not-applicable)",
        "explanation": "Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (No, open, not-applicable)."
      },
      "skills": [
        "Open-Sentence Binding and Truth Eligibility"
      ],
      "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": "古典整數語境；語句x≤-2，assignment=null（未綁定）。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。",
        "en": "In the classical integer context, consider x≤-2 with assignment=null (unbound). Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "classical integer formulas with explicit binding/data",
      "after": "a truth verdict is not a delivered general proof object",
      "relation": "implies",
      "rule_id": "proposition_truth_and_proof_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "假陳述仍是命題；這裡提供有限代入與比較見證，不把真值標記升格為一般形式證明或任意推理規則的完備性。",
        "en": "A false closed statement is still a proposition.These finite substitutions and comparison witnesses do not promote a truth label to a general formal proof or completeness of arbitrary inference rules."
      },
      "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": "formula x≤-2;assignment=null",
      "after": "no bound integer for x;leave truth and numeric difference inapplicable",
      "relation": "implies",
      "rule_id": "unbound_formula_not_false",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此協議沒有給x綁定，不能賦一個固定真假或偷偷補上全稱量詞；未綁定不是假，null也不是0。",
        "en": "No x binding is supplied in this protocol;do not assign a fixed truth value or silently add a universal quantifier.Unbound is not False,and null is not0."
      },
      "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": [
      false,
      "未封閉",
      "不適用"
    ],
    "binding_state": "unbound_open_formula"
  },
  "conclusion": {
    "canonical_value": "(否, 未封閉, 不適用)",
    "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-20261012-r03-proposition-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261012-r03-proposition-foundations/questions.jsonl",
      "batch_sha256": "5b0a0faf5bf6b380d452a972543ab58c4753045b86dd5aa697810565dc73b8e9",
      "original_line": 26,
      "original_line_sha256": "37898bc546588bf58912a371e6f781ef6b26dce8c9b892873361232a4a583db7",
      "question_compact_sha256": "0213658fad3206e7b0ee2063567e34f2c15c4d2cf79a36b1187d7704aa0d11e5",
      "english_row_compact_sha256": "0a3aec17c1eb78e3cd3b2d2c19a4118ab0663bc43105650e06316b6ca1fac6aa",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261012-r03-proposition-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261012-r03-proposition-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-proposition": {
        "sha256": "77a5e0d8403fa47d1ee57431b11f6fc52bfdde80a575db9dac3589fb0b9e8510",
        "english_sha256": "3d649ce5ebed9f16e8aea38557e60c43c771430b81bdc1be7533001a956f0608",
        "original_path": "public/data/mko/proposition.json",
        "english_path": "public/data/i18n/editions/mko-proposition.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "aeab777a3897ae2c01315b86b5e9b9a43ffb59b075efae44fd95d447123d32f1",
    "original_candidate_sha256": "f7b979d81aef8dff8b5a7cbe7b7d6557c3584392e4a5f30e60fd3ed3fbe8ae26",
    "original_candidate_line_sha256": "f7b979d81aef8dff8b5a7cbe7b7d6557c3584392e4a5f30e60fd3ed3fbe8ae26"
  },
  "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-20261026-005-p11",
    "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": "0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f",
      "question_first_witness_sha256": "0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc"
    },
    "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": "af46aa3ec3c23d332c217ee03e43ea38d8ebaab8b3fb52a8875f7434fbc56848",
      "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."
  }
}