{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261028-r03-real-banach-space-0075",
  "pilot_number": "R0025-13450",
  "number": 13450,
  "family_id": "qf-mod28-banach-distance-evaluate",
  "title": {
    "zh-Hant": "第 13450 題：實Banach空間",
    "en": "Question 13450: A Named-Norm Distance Is Not a Completeness Proof"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261028-r03-real-banach-space-0075",
    "batch_id": "batch-20261014-advance-20261028-r03-real-banach-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-real-banach-space"
    ],
    "title_zh": "實Banach空間",
    "stem_zh": "載體norm誘導metric與完備證書必須描述同一空間。操作=banach-distance-evaluate；資料={\"norm_name\":\"l1\",\"left\":[12,0,1],\"right\":[-12,-3,-2]}。請依 [norm_name,left,right,difference,distance,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30, distance_evaluation_not_completeness_proof)",
      "explanation_zh": "Wave4第二層核對=(l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30, distance_evaluation_not_completeness_proof)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；有限維實賦範空間使用命名全稱完備定理；抽樣Cauchy序列不足且不推出Hilbert。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "實Banach空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod28-banach-distance-evaluate",
      "seed": 202610283075,
      "parameters": {
        "norm_name": "l1",
        "left": [
          12,
          0,
          1
        ],
        "right": [
          -12,
          -3,
          -2
        ],
        "domain": "Named_finite_dimensional_real_Banach_space",
        "operation": "banach-distance-evaluate",
        "semantic_template": "qf-mod28-banach-distance-evaluate/{\"left\":[12,0,1],\"norm_name\":\"l1\",\"right\":[-12,-3,-2]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式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-20261028-r03-real-banach-space-0075",
    "batch_id": "batch-20261014-advance-20261028-r03-real-banach-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a088b492960aca7294c374e1798187fd389a9b1f09c8a759b78f6ae190500d1e",
      "family_id": "qf-mod28-banach-distance-evaluate",
      "seed": 202610283075,
      "semantic_template": "qf-mod28-banach-distance-evaluate/{\"left\":[12,0,1],\"norm_name\":\"l1\",\"right\":[-12,-3,-2]}"
    },
    "fields": {
      "title": "A Named-Norm Distance Is Not a Completeness Proof",
      "stem": "Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[12, 0, 1], right=[-12, -3, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).",
      "answer": {
        "kind": "expression",
        "value": "(l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30, distance_evaluation_not_completeness_proof)",
        "explanation": "Apply the fixed named norm to the signed difference. This exact evaluation does not itself quantify over Cauchy sequences. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30, distance_evaluation_not_completeness_proof)."
      },
      "skills": [
        "A Named-Norm Distance Is Not a Completeness Proof"
      ],
      "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": "載體norm誘導metric與完備證書必須描述同一空間。操作=banach-distance-evaluate；資料={\"norm_name\":\"l1\",\"left\":[12,0,1],\"right\":[-12,-3,-2]}。請依 [norm_name,left,right,difference,distance,boundary] 欄序回答。",
        "en": "Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[12, 0, 1], right=[-12, -3, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Banach coordinate distance difference coordinate0 := sub(12, -12)",
      "after": "Banach coordinate distance difference coordinate0 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。",
        "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Banach coordinate distance difference coordinate0，不以小數近似替代。",
          "en": "Recompute Banach coordinate distance difference coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "12",
          "-12"
        ],
        "output": "24",
        "label": "Banach coordinate distance difference coordinate0"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Banach coordinate distance difference coordinate1 := sub(0, -3)",
      "after": "Banach coordinate distance difference coordinate1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。",
        "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Banach coordinate distance difference coordinate1，不以小數近似替代。",
          "en": "Recompute Banach coordinate distance difference coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-3"
        ],
        "output": "3",
        "label": "Banach coordinate distance difference coordinate1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Banach coordinate distance difference coordinate2 := sub(1, -2)",
      "after": "Banach coordinate distance difference coordinate2 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。",
        "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Banach coordinate distance difference coordinate2，不以小數近似替代。",
          "en": "Recompute Banach coordinate distance difference coordinate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "3",
        "label": "Banach coordinate distance difference coordinate2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Banach coordinate distance:norm=l1;left=[12, 0, 1];right=[-12, -3, -2]",
      "after": "signed left-right=[24, 3, 3];use same named norm",
      "relation": "implies",
      "rule_id": "norm_distance_same_space_and_direction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在同一實座標空間以同一指定範數取差的範數；差可超過原輸入座標界，不能錯當非法新輸入而截斷。",
        "en": "Use the same stated norm on the same real coordinate space.The derived difference can exceed original input coordinate bounds and must not be clipped or rejected as a new raw input."
      },
      "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": "Banach coordinate distance:coordinate=0;signed value=24",
      "after": "absolute value=24",
      "relation": "implies",
      "rule_id": "named_norm_coordinate_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取完整座標絕對值，負座標的符號保留在原向量，不直接拿帶號最大值。",
        "en": "Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum."
      },
      "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": "Banach coordinate distance:coordinate=1;signed value=3",
      "after": "absolute value=3",
      "relation": "implies",
      "rule_id": "named_norm_coordinate_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取完整座標絕對值，負座標的符號保留在原向量，不直接拿帶號最大值。",
        "en": "Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum."
      },
      "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": "Banach coordinate distance:coordinate=2;signed value=3",
      "after": "absolute value=3",
      "relation": "implies",
      "rule_id": "named_norm_coordinate_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取完整座標絕對值，負座標的符號保留在原向量，不直接拿帶號最大值。",
        "en": "Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum."
      },
      "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": "Banach coordinate distance l1 sum := sum(24, 3, 3)",
      "after": "Banach coordinate distance l1 sum = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。",
        "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Banach coordinate distance l1 sum，不以小數近似替代。",
          "en": "Recompute Banach coordinate distance l1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "3",
          "3"
        ],
        "output": "30",
        "label": "Banach coordinate distance l1 sum"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "fixed carrier R^3;named norm=l1;distance=30",
      "after": "distance_evaluation_not_completeness_proof",
      "relation": "implies",
      "rule_id": "Banach_distance_evaluation_not_all_Cauchy_claim",
      "conditions": [],
      "explanation": {
        "zh-Hant": "算出一個距離沒有量化每個Cauchy序列，不以這個數值另證完備性。",
        "en": "One distance evaluation quantifies over no Cauchy sequences and is not itself a completeness proof."
      },
      "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": [
      "l1",
      [
        12,
        0,
        1
      ],
      [
        -12,
        -3,
        -2
      ],
      [
        24,
        3,
        3
      ],
      30,
      "distance_evaluation_not_completeness_proof"
    ]
  },
  "conclusion": {
    "canonical_value": "(l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30, distance_evaluation_not_completeness_proof)",
    "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": {
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        "original_path": "public/data/mko/real-banach-space.json",
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    "original_candidate_sha256": "9f83860c2bf8c32059616178c8bee265abc050a21055c9d6fcc7ff3cdabe365f",
    "original_candidate_line_sha256": "9f83860c2bf8c32059616178c8bee265abc050a21055c9d6fcc7ff3cdabe365f"
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      "formalization": "not_formalized",
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    "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-20261026-007-p26",
    "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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    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}