{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261005-advance-20261006-r01-order-relation-0059",
  "pilot_number": "R0067-34034",
  "number": 34034,
  "family_id": "qf-order-closed-interval",
  "title": {
    "zh-Hant": "第 34034 題：閉區間的含端點判定",
    "en": "Question 34034: Membership Including Closed Endpoints"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261005-advance-20261006-r01-order-relation-0059",
    "batch_id": "batch-20261005-advance-20261006-r01-order-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-order-relation"
    ],
    "title_zh": "閉區間的含端點判定",
    "stem_zh": "依實數順序，正分母有理數 11/1 是否屬於閉區間 [-8,10]？閉區間包含兩端點，允許單點區間。",
    "answer": {
      "kind": "short_text",
      "value": "否",
      "explanation_zh": "閉區間[-8,10]；值=11/1。將兩個非嚴格不等式同乘正分母，兩側都成立才屬於；端點相等仍按閉區間慣例判定。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "閉區間的含端點判定"
      ]
    },
    "variant": {
      "family_id": "qf-order-closed-interval",
      "seed": 202610061059,
      "parameters": {
        "p": 11,
        "q": 1,
        "lower": -8,
        "upper": 10,
        "operation": "closed_interval",
        "semantic_template": "qf-order-closed-interval/{\"inputs\":[\"11/1\",\"-8\",\"10\"],\"operation\":\"closed_interval\"}"
      }
    },
    "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": "2026-10-05",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261005-advance-20261006-r01-order-relation-0059",
    "batch_id": "batch-20261005-advance-20261006-r01-order-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2a010bf8b7c3e480935585c9cd3d1aa24ce11d8b4bc1b0c513784deb3bf4f300",
      "family_id": "qf-order-closed-interval",
      "seed": 202610061059,
      "semantic_template": "qf-order-closed-interval/{\"inputs\":[\"11/1\",\"-8\",\"10\"],\"operation\":\"closed_interval\"}"
    },
    "fields": {
      "title": "Membership Including Closed Endpoints",
      "stem": "Under the usual real order, is (11)/1 in the closed interval [-8, 10]? The denominator is positive and both endpoints are included.",
      "answer": {
        "kind": "short_text",
        "value": "No",
        "explanation": "Check -8·1≤11≤10·1 exactly, obtaining No. A singleton closed interval still contains its endpoint."
      },
      "skills": [
        "Membership Including Closed Endpoints"
      ],
      "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": "依實數順序，正分母有理數 11/1 是否屬於閉區間 [-8,10]？閉區間包含兩端點，允許單點區間。",
        "en": "Under the usual real order, is (11)/1 in the closed interval [-8, 10]? The denominator is positive and both endpoints are included."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-order-closed-interval, [lower, operation, p, q, upper]]",
      "after": "derived = [closed_interval, no-invented-domain]",
      "relation": "implies",
      "rule_id": "R67_SOURCE_specific_keys_operation_domain_and_numeric_bank",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本bank核keys與operation/domain：legacy為各表underscore，MOD50用local hyphen；BASE參數較寬不加未知欄限制。Number.isInteger與SAFE兩種來源界分開，原JSON數字表示仍完整保存。",
        "en": "Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。",
          "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Interval ORIGINAL lower times positive denominator := mul(-8, 1)",
      "after": "Interval ORIGINAL lower times positive denominator = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。",
        "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Interval ORIGINAL lower times positive denominator，不以小數近似替代。",
          "en": "Recompute Interval ORIGINAL lower times positive denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-8",
          "1"
        ],
        "output": "-8",
        "label": "Interval ORIGINAL lower times positive denominator"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Interval ORIGINAL upper times positive denominator := mul(10, 1)",
      "after": "Interval ORIGINAL upper times positive denominator = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。",
        "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Interval ORIGINAL upper times positive denominator，不以小數近似替代。",
          "en": "Recompute Interval ORIGINAL upper times positive denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "1"
        ],
        "output": "10",
        "label": "Interval ORIGINAL upper times positive denominator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [11, 1, -8, 10]",
      "after": "derived = [-8, 11, 10, 是, 否, 否]",
      "relation": "implies",
      "rule_id": "Order_CLOSED_interval_BOTH_inclusive_inequalities",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分母正時lower≤p/q≤upper等價於lower*q≤p≤upper*q，兩端都含等號。lower=upper合法但只包含該值；不能把閉區間當開區間或默默排序壞端點。",
        "en": "For a positive denominator,lower≤p/q≤upper is equivalent to lower*q≤p≤upper*q.BOTH endpoints are included.A singleton is valid but contains only its endpoint.Do not use open inequalities or silently sort reversed endpoints."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。",
          "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "否"
  },
  "conclusion": {
    "canonical_value": "否",
    "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-20261005-advance-20261006-r01-order-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261005-advance-20261006-r01-order-relation-foundations/questions.jsonl",
      "batch_sha256": "7445d3cd6c8f4cc77448d51656238ce18229a8b5084f76112e419933e63d33a8",
      "original_line": 59,
      "original_line_sha256": "06518a0df33d287547cf1181d72a64dd196fa7c9c0baafff0a9bb92f9f9d335e",
      "question_compact_sha256": "2a010bf8b7c3e480935585c9cd3d1aa24ce11d8b4bc1b0c513784deb3bf4f300",
      "english_row_compact_sha256": "afc12df7834515ab2b883f0ea7bd186e2b85694ca5e57142e3b0f66f70a96038",
      "english_manifest_path": "public/data/i18n/editions/batch-20261005-advance-20261006-r01-order-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261005-advance-20261006-r01-order-relation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-order-relation": {
        "sha256": "95554189da3a451f8a69edd139e154fd9254524119837ea1643487a5646a2dc3",
        "english_sha256": "3bb53f0ce66755124a9e09f8d185cdebfbb4e6f8cc93a7733a5c08ceba6bdcd6",
        "original_path": "public/data/mko/order-relation.json",
        "english_path": "public/data/i18n/editions/mko-order-relation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bb156cec7bdb6baf2497762702fc01f94fd1db55a5926bc575b12ff9237a4de",
    "original_candidate_sha256": "61843d83b4ac318ae43477f2bce04c80930378ee088a804f18aa221ee3da86e2",
    "original_candidate_line_sha256": "61843d83b4ac318ae43477f2bce04c80930378ee088a804f18aa221ee3da86e2"
  },
  "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-20261026-012-p68",
    "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": "c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c",
      "question_first_witness_sha256": "12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "bca71e82a3b603522a7c8b63fcbc81f6ac763288e5b122c1663da40e54865d65",
      "arithmetic_steps": 2
    },
    "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."
  }
}