{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261012-r03-proposition-0028",
  "pilot_number": "R0010-05753",
  "number": 5753,
  "family_id": "qf-density12-proposition-open-binding",
  "title": {
    "zh-Hant": "第 5753 題：自由變數語句的封閉資格",
    "en": "Question 5753: Open-Sentence Binding and Truth Eligibility"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261012-r03-proposition-0028",
    "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≤0，assignment=-10。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": "(是, 真, -10)",
      "explanation_zh": "封閉資格與代入真假=(是, 真, -10)。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": 202610123028,
      "parameters": {
        "threshold": 0,
        "assignment": -10,
        "operation": "density12_proposition_open_binding",
        "domain": "integer_open_sentence_binding",
        "semantic_template": "qf-density12-proposition-open-binding/\"-10\""
      }
    },
    "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-0028",
    "batch_id": "batch-20261008-advance-20261012-r03-proposition-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c271b9b3162d7892c309c53bd1826ec3defe8204899b97a2b9abce98c5abbcb2",
      "family_id": "qf-density12-proposition-open-binding",
      "seed": 202610123028,
      "semantic_template": "qf-density12-proposition-open-binding/\"-10\""
    },
    "fields": {
      "title": "Open-Sentence Binding and Truth Eligibility",
      "stem": "In the classical integer context, consider x≤0 with assignment=-10. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).",
      "answer": {
        "kind": "expression",
        "value": "(Yes, True, -10)",
        "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 (Yes, True, -10)."
      },
      "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≤0，assignment=-10。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≤0 with assignment=-10. 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": "bound integer comparison difference := sub(-10, 0)",
      "after": "bound integer comparison difference = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有整數綁定後才代入，差為x−門檻。",
        "en": "Substitute only after receiving an integer binding;the difference is x minus the threshold."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bound integer comparison difference，不以小數近似替代。",
          "en": "Recompute bound integer comparison difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-10",
          "0"
        ],
        "output": "-10",
        "label": "bound integer comparison difference"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "x=-10;threshold=0;x−threshold=-10",
      "after": "bound statement truth=真",
      "relation": "implies",
      "rule_id": "bound_integer_formula_verdict",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已綁定的0也是合法值；差≤0恰對應x≤門檻，不用Python或程式語言的真假性代替數學綁定。",
        "en": "A bound0 is a valid value too.The condition difference≤0 is exactly x≤threshold;programming-language truthiness is not mathematical binding."
      },
      "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": [
      true,
      true,
      "-10"
    ],
    "binding_state": "bound_integer_formula"
  },
  "conclusion": {
    "canonical_value": "(是, 真, -10)",
    "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": 28,
      "original_line_sha256": "296dd96b2170e19509be5c7d7d9b5b849b1ce6bf2cb65b504aeb3798b8da6c4c",
      "question_compact_sha256": "c271b9b3162d7892c309c53bd1826ec3defe8204899b97a2b9abce98c5abbcb2",
      "english_row_compact_sha256": "21bf1ffc2dd07b5062cfb90063abb133750c025488bd585c7610dbbb95938a8d",
      "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": "7f93fdaaf50fba2cb5ed5d95e3f2fb2ea819ce29dc30d6a0772c4d4a4c60847b",
    "original_candidate_line_sha256": "7f93fdaaf50fba2cb5ed5d95e3f2fb2ea819ce29dc30d6a0772c4d4a4c60847b"
  },
  "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": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4809da64f13f6aee681de918b92e9e82aa10d91403a21995699503d550581504",
      "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."
  }
}