{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261017-r04-order-relation-0072",
  "pilot_number": "R0015-08272",
  "number": 8272,
  "family_id": "qf-density17-order-three-membership-slacks",
  "title": {
    "zh-Hant": "第 8272 題：閉區間會員與兩側鬆弛量",
    "en": "Question 8272: Three Closed-Interval Membership Tests and Signed Slacks"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261017-r04-order-relation-0072",
    "batch_id": "batch-20261012-advance-20261017-r04-order-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-order-relation"
    ],
    "title_zh": "閉區間會員與兩側鬆弛量",
    "stem_zh": "R通常全序，原分母為正。閉區間=[(-21)/(3), (21)/(2)]，有序三查詢=[(0)/(1), (-21)/(3), (22)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。",
    "answer": {
      "kind": "expression",
      "value": "([-7/1, 21/2], [0/1, -7/1, 11/1], [是, 是, 否], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])",
      "explanation_zh": "離散結構核對=([-7/1, 21/2], [0/1, -7/1, 11/1], [是, 是, 否], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "閉區間會員與兩側鬆弛量"
      ]
    },
    "variant": {
      "family_id": "qf-density17-order-three-membership-slacks",
      "seed": 202610174072,
      "parameters": {
        "l": [
          -21,
          3
        ],
        "u": [
          21,
          2
        ],
        "queries": [
          [
            0,
            1
          ],
          [
            -21,
            3
          ],
          [
            22,
            2
          ]
        ],
        "domain": "R_positive_denominator_order",
        "operation": "order-three-membership-slacks",
        "semantic_template": "qf-density17-order-three-membership-slacks/{\"l\":\"-7/1\",\"queries\":[\"0/1\",\"-7/1\",\"11/1\"],\"u\":\"21/2\"}"
      }
    },
    "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": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261017-r04-order-relation-0072",
    "batch_id": "batch-20261012-advance-20261017-r04-order-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2731f3709bd7b413ce23d205afada08976f5eb679e61d1ed4a731feeb0b3d602",
      "family_id": "qf-density17-order-three-membership-slacks",
      "seed": 202610174072,
      "semantic_template": "qf-density17-order-three-membership-slacks/{\"l\":\"-7/1\",\"queries\":[\"0/1\",\"-7/1\",\"11/1\"],\"u\":\"21/2\"}"
    },
    "fields": {
      "title": "Three Closed-Interval Membership Tests and Signed Slacks",
      "stem": "Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-21/3, 21/2], queries=[0/1, -21/3, 22/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).",
      "answer": {
        "kind": "expression",
        "value": "([-7/1, 21/2], [0/1, -7/1, 11/1], [Yes, Yes, No], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])",
        "explanation": "Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/1, 21/2], [0/1, -7/1, 11/1], [Yes, Yes, No], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])."
      },
      "skills": [
        "Three Closed-Interval Membership Tests and Signed Slacks"
      ],
      "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": "R通常全序，原分母為正。閉區間=[(-21)/(3), (21)/(2)]，有序三查詢=[(0)/(1), (-21)/(3), (22)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。",
        "en": "Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-21/3, 21/2], queries=[0/1, -21/3, 22/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "usual total order on R;all original denominators positive",
      "after": "closed endpoints included;negative scaling reverses;zero collapses",
      "relation": "implies",
      "rule_id": "real_rational_order_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此題是通常實數全序的精確有理切片，不外推複數或任意偏序；原分母必正，不能先正規化負分母來略過原規格。",
        "en": "These exact rationals use the usual real total order,not complex numbers or an arbitrary partial order.Original denominators must be positive;normalizing a negative input denominator does not waive that contract."
      },
      "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": "query0 x-lower := sub(0, -7)",
      "after": "query0 x-lower = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 x-lower，不以小數近似替代。",
          "en": "Recompute query0 x-lower with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-7"
        ],
        "output": "7",
        "label": "query0 x-lower"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "query0 upper-x := sub(21/2, 0)",
      "after": "query0 upper-x = 21/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 upper-x，不以小數近似替代。",
          "en": "Recompute query0 upper-x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "21/2",
          "0"
        ],
        "output": "21/2",
        "label": "query0 upper-x"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "query0=0/1;signed slacks=[7/1, 21/2]",
      "after": "both nonnegative=是",
      "relation": "implies",
      "rule_id": "closed_interval_both_signed_slacks",
      "conditions": [],
      "explanation": {
        "zh-Hant": "必須兩個有號差同時非負；端點差為0仍在閉區間內，不把負差取絕對值。",
        "en": "Both signed slacks must be nonnegative.Zero includes a closed endpoint;do not take absolute values of negative slacks."
      },
      "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": "query1 x-lower := sub(-7, -7)",
      "after": "query1 x-lower = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 x-lower，不以小數近似替代。",
          "en": "Recompute query1 x-lower with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-7",
          "-7"
        ],
        "output": "0",
        "label": "query1 x-lower"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "query1 upper-x := sub(21/2, -7)",
      "after": "query1 upper-x = 35/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 upper-x，不以小數近似替代。",
          "en": "Recompute query1 upper-x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "21/2",
          "-7"
        ],
        "output": "35/2",
        "label": "query1 upper-x"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "query1=-7/1;signed slacks=[0/1, 35/2]",
      "after": "both nonnegative=是",
      "relation": "implies",
      "rule_id": "closed_interval_both_signed_slacks",
      "conditions": [],
      "explanation": {
        "zh-Hant": "必須兩個有號差同時非負；端點差為0仍在閉區間內，不把負差取絕對值。",
        "en": "Both signed slacks must be nonnegative.Zero includes a closed endpoint;do not take absolute values of negative slacks."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "query2 x-lower := sub(11, -7)",
      "after": "query2 x-lower = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 x-lower，不以小數近似替代。",
          "en": "Recompute query2 x-lower with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "-7"
        ],
        "output": "18",
        "label": "query2 x-lower"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "query2 upper-x := sub(21/2, 11)",
      "after": "query2 upper-x = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 upper-x，不以小數近似替代。",
          "en": "Recompute query2 upper-x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "21/2",
          "11"
        ],
        "output": "-1/2",
        "label": "query2 upper-x"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "query2=11/1;signed slacks=[18/1, -1/2]",
      "after": "both nonnegative=否",
      "relation": "implies",
      "rule_id": "closed_interval_both_signed_slacks",
      "conditions": [],
      "explanation": {
        "zh-Hant": "必須兩個有號差同時非負；端點差為0仍在閉區間內，不把負差取絕對值。",
        "en": "Both signed slacks must be nonnegative.Zero includes a closed endpoint;do not take absolute values of negative slacks."
      },
      "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": [
      [
        "-7/1",
        "21/2"
      ],
      [
        "0/1",
        "-7/1",
        "11/1"
      ],
      [
        true,
        true,
        false
      ],
      [
        [
          "7/1",
          "21/2"
        ],
        [
          "0/1",
          "35/2"
        ],
        [
          "18/1",
          "-1/2"
        ]
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-7/1, 21/2], [0/1, -7/1, 11/1], [是, 是, 否], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])",
    "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-20261012-advance-20261017-r04-order-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261017-r04-order-relation-foundations/questions.jsonl",
      "batch_sha256": "6bee78c2e753258a82042d9051a2179c845945ab84af56cbb9e83a8e573cf599",
      "original_line": 72,
      "original_line_sha256": "ae959208066faf1bbdc4ae6ab4d7cbcda6571273ca1b9396b5e73296dc1b6c2b",
      "question_compact_sha256": "2731f3709bd7b413ce23d205afada08976f5eb679e61d1ed4a731feeb0b3d602",
      "english_row_compact_sha256": "26c187a946827b9b02682ffb5dc378e89139ee58d7e1e9939068860aa080442f",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261017-r04-order-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261017-r04-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": "ca1078dc467a0f3769865919aadc46c87712d098f1dfbde90f787471e22bcfec",
    "original_candidate_sha256": "7f316514666ce05dbd05a1cc3637fba426554a844efa092a6c0f753b13b9188e",
    "original_candidate_line_sha256": "7f316514666ce05dbd05a1cc3637fba426554a844efa092a6c0f753b13b9188e"
  },
  "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-p16",
    "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": "6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18",
      "question_first_witness_sha256": "e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4e4285a959d6e55c3801d87727979f498b64484c9692e3a0930dff70397b39e4",
      "arithmetic_steps": 6
    },
    "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."
  }
}