{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261107-r02-supremum-and-infimum-0089",
  "pilot_number": "R0040-20564",
  "number": 20564,
  "family_id": "qf-mod38-supinf-open-interval-nonattainment",
  "title": {
    "zh-Hant": "第 20564 題：上確界與下確界",
    "en": "Question 20564: The Bounds of an Open Interval Are Not Its Maximum or Minimum"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261107-r02-supremum-and-infimum-0089",
    "batch_id": "batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-supremum-and-infimum"
    ],
    "title_zh": "上確界與下確界",
    "stem_zh": "非空且相應有界的實值集合或明示有界開區間。操作=supinf-open-interval-nonattainment；資料={\"left\":[13,14],\"right\":[15,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(13/14, 15/14, 15/14, 13/14, 否, 否, supremum_and_infimum_not_attained)",
      "explanation_zh": "Wave6根層核對=(13/14, 15/14, 15/14, 13/14, 否, 否, supremum_and_infimum_not_attained)；依完整明示資料獨立重算；sup同時是上界與最小上界，inf對偶；兩項責任不可省略；sup與inf不必屬於集合，只有取到時才是maximum或minimum；空集與無界集另處理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "上確界與下確界"
      ]
    },
    "variant": {
      "family_id": "qf-mod38-supinf-open-interval-nonattainment",
      "seed": 202611072089,
      "parameters": {
        "left": [
          13,
          14
        ],
        "right": [
          15,
          14
        ],
        "domain": "Exact_bounded_real_set_extrema",
        "operation": "supinf-open-interval-nonattainment",
        "semantic_template": "qf-mod38-supinf-open-interval-nonattainment/{\"left\":[13,14],\"right\":[15,14]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261107-r02-supremum-and-infimum-0089",
    "batch_id": "batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2ba7e97cf022d2205c37b24d190c698ad1407f1f2c7230de84a9dd44bd07a87c",
      "family_id": "qf-mod38-supinf-open-interval-nonattainment",
      "seed": 202611072089,
      "semantic_template": "qf-mod38-supinf-open-interval-nonattainment/{\"left\":[13,14],\"right\":[15,14]}"
    },
    "fields": {
      "title": "The Bounds of an Open Interval Are Not Its Maximum or Minimum",
      "stem": "Use the entire nonempty bounded real open interval (13/14, 15/14), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(13/14, 15/14, 15/14, 13/14, No, No, supremum_and_infimum_not_attained)",
        "explanation": "The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (13/14, 15/14, 15/14, 13/14, No, No, supremum_and_infimum_not_attained)."
      },
      "skills": [
        "The Bounds of an Open Interval Are Not Its Maximum or Minimum"
      ],
      "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": "非空且相應有界的實值集合或明示有界開區間。操作=supinf-open-interval-nonattainment；資料={\"left\":[13,14],\"right\":[15,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the entire nonempty bounded real open interval (13/14, 15/14), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=supinf-open-interval-nonattainment;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[left, right, supremum, infimum, supremum_attained, infimum_attained, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。",
        "en": "Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope."
      },
      "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": "supinf WHOLE open interval positive width := sub(15/14, 13/14)",
      "after": "supinf WHOLE open interval positive width = 1/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 supinf WHOLE open interval positive width，不以小數近似替代。",
          "en": "Recompute supinf WHOLE open interval positive width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "15/14",
          "13/14"
        ],
        "output": "1/7",
        "label": "supinf WHOLE open interval positive width"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "supinf interval midpoint numerator := sum(13/14, 15/14)",
      "after": "supinf interval midpoint numerator = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 supinf interval midpoint numerator，不以小數近似替代。",
          "en": "Recompute supinf interval midpoint numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "13/14",
          "15/14"
        ],
        "output": "2",
        "label": "supinf interval midpoint numerator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "supinf one midpoint nonemptiness witness := div(2, 2)",
      "after": "supinf one midpoint nonemptiness witness = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 supinf one midpoint nonemptiness witness，不以小數近似替代。",
          "en": "Recompute supinf one midpoint nonemptiness witness with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "2"
        ],
        "output": "1",
        "label": "supinf one midpoint nonemptiness witness"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "WHOLE original real open interval(13/14,15/14);positive width=1/7;one displayed midpoint=1",
      "after": "for EVERY real x in this WHOLE interval,13/14<x<15/14;right=15/14 is an upper bound,left=13/14 is a lower bound;midpoint only witnesses nonemptiness",
      "relation": "implies",
      "rule_id": "supinf_infinite_interval_bounds_not_finite_sample_extrema",
      "conditions": [],
      "explanation": {
        "zh-Hant": "區間包含全部滿足嚴格不等式的實數，不把midpoint當完整集合或由一點猜sup/inf；bound理由是任意區間內x的定義不等式。",
        "en": "The interval contains every real satisfying its strict inequalities,not just the midpoint.Bounds follow from the defining inequalities for every interval point,not from extrema of a sample."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "WHOLE interval left L=13/14,right R=15/14;consider ANY real proposed upper bound b<R or lower bound c>L",
      "after": "x=(max(L,b)+R)/2 lies in the interval and x>b,so b is NOT upper;y=(L+min(R,c))/2 lies in the interval and y<c,so c is NOT lower;therefore sup=R and inf=L",
      "relation": "implies",
      "rule_id": "supinf_open_interval_all_real_candidate_leastness_greatestness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "b<R時max(L,b)<R，兩者midpoint嚴格介於其間而超過b；c>L時同理用min(R,c)得到低於c的區間點。這對任意實candidate有效，不限有理取樣或一串顯示數值。",
        "en": "For b<R,max(L,b)<R,so their midpoint is an interval point greater than b.The analogous midpoint using min(R,c) lies below c for any c>L.These arguments cover arbitrary real candidates,not only rational samples or a displayed sequence."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "original interval conditions L<x<R with L=13/14,R=15/14",
      "after": "NEITHER endpoint satisfies both strict inequalities;supremum attained=否,infimum attained=否;for any interior point a midpoint toward each endpoint gives a larger/smaller interior point",
      "relation": "implies",
      "rule_id": "supinf_bounds_not_members_no_maximum_or_minimum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "端點不在開區間；任意內點往兩端各取midpoint還可得到更大／更小內點，故沒有maximum或minimum。sup/inf存在不等於取到。",
        "en": "Neither endpoint belongs to the open interval.From any interior point,midpoints toward the endpoints give larger/smaller interior points,so no maximum or minimum exists.Existence of sup/inf is different from attainment."
      },
      "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": [
      "13/14",
      "15/14",
      "15/14",
      "13/14",
      false,
      false,
      "supremum_and_infimum_not_attained"
    ]
  },
  "conclusion": {
    "canonical_value": "(13/14, 15/14, 15/14, 13/14, 否, 否, supremum_and_infimum_not_attained)",
    "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."
  },
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    "original_candidate_line_sha256": "78cb27165f476ef38317b90204670140403967b1c711dde92cf2dfbde7e3de19",
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      "formalization": "not_formalized",
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      "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."
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    "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": {
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    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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}