{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r02-real-normed-vector-space-0093",
  "pilot_number": "R0024-12618",
  "number": 12618,
  "family_id": "qf-mod26-norm-induced-distance",
  "title": {
    "zh-Hant": "第 12618 題：實賦範向量空間",
    "en": "Question 12618: The Norm of an Ordered Vector Difference"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r02-real-normed-vector-space-0093",
    "batch_id": "batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-real-normed-vector-space"
    ],
    "title_zh": "實賦範向量空間",
    "stem_zh": "只使用明示l1或linfinity精確公式。操作=norm-induced-distance；資料={\"norm_name\":\"linfinity\",\"left\":[5,0,2],\"right\":[-5,1,-1]}。請依 [norm_name,left,right,difference,distance] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10)",
      "explanation_zh": "結構深度核對=(linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；命名公式提供全稱代數證書；有限點抽樣不能證範數公理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "實賦範向量空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-norm-induced-distance",
      "seed": 202610262093,
      "parameters": {
        "norm_name": "linfinity",
        "left": [
          5,
          0,
          2
        ],
        "right": [
          -5,
          1,
          -1
        ],
        "domain": "Named_exact_norms_on_Rn",
        "operation": "norm-induced-distance",
        "semantic_template": "qf-mod26-norm-induced-distance/{\"left\":[5,0,2],\"norm_name\":\"linfinity\",\"right\":[-5,1,-1]}"
      }
    },
    "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、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261026-r02-real-normed-vector-space-0093",
    "batch_id": "batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "40960d8b5bb350e6f5bac09a732ecbd1cb876a5b3f7d8bc014b87461a4541bad",
      "family_id": "qf-mod26-norm-induced-distance",
      "seed": 202610262093,
      "semantic_template": "qf-mod26-norm-induced-distance/{\"left\":[5,0,2],\"norm_name\":\"linfinity\",\"right\":[-5,1,-1]}"
    },
    "fields": {
      "title": "The Norm of an Ordered Vector Difference",
      "stem": "Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[5, 0, 2], right=[-5, 1, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).",
      "answer": {
        "kind": "expression",
        "value": "(linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10)",
        "explanation": "Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10)."
      },
      "skills": [
        "The Norm of an Ordered Vector Difference"
      ],
      "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": "只使用明示l1或linfinity精確公式。操作=norm-induced-distance；資料={\"norm_name\":\"linfinity\",\"left\":[5,0,2],\"right\":[-5,1,-1]}。請依 [norm_name,left,right,difference,distance] 欄序回答。",
        "en": "Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[5, 0, 2], right=[-5, 1, -1] with equal dimension. Give (norm_name,left,right,left−right,distance)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "distance coordinate0 left-minus-right := sub(5, -5)",
      "after": "distance coordinate0 left-minus-right = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 distance coordinate0 left-minus-right，不以小數近似替代。",
          "en": "Recompute distance coordinate0 left-minus-right with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "-5"
        ],
        "output": "10",
        "label": "distance coordinate0 left-minus-right"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "distance coordinate1 left-minus-right := sub(0, 1)",
      "after": "distance coordinate1 left-minus-right = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 distance coordinate1 left-minus-right，不以小數近似替代。",
          "en": "Recompute distance coordinate1 left-minus-right with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "1"
        ],
        "output": "-1",
        "label": "distance coordinate1 left-minus-right"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "distance coordinate2 left-minus-right := sub(2, -1)",
      "after": "distance coordinate2 left-minus-right = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 distance coordinate2 left-minus-right，不以小數近似替代。",
          "en": "Recompute distance coordinate2 left-minus-right with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "3",
        "label": "distance coordinate2 left-minus-right"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "named certified norm=linfinity;left=[5, 0, 2];right=[-5, 1, -1]",
      "after": "ordered difference=[10, -1, 3];distance=norm(left-right)",
      "relation": "implies",
      "rule_id": "fixed_norm_induced_distance_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在同一座標空間用同一命名範數；保留差的方向。這不說所有metric皆由norm誘導，也不額外宣稱完備或內積。",
        "en": "Use the same named norm on the same coordinate space and retain the difference direction.This does not assert every metric is norm-induced,or add completeness or inner-product claims."
      },
      "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": "induced distance:coordinate=0;signed value=10",
      "after": "absolute value=10",
      "relation": "implies",
      "rule_id": "named_norm_each_coordinate_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取每個完整座標的絕對值，負值不直接加入或當作最大值；保留原向量符號。",
        "en": "Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector."
      },
      "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": "induced distance:coordinate=1;signed value=-1",
      "after": "absolute value=1",
      "relation": "implies",
      "rule_id": "named_norm_each_coordinate_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取每個完整座標的絕對值，負值不直接加入或當作最大值；保留原向量符號。",
        "en": "Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector."
      },
      "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": "induced distance:coordinate=2;signed value=3",
      "after": "absolute value=3",
      "relation": "implies",
      "rule_id": "named_norm_each_coordinate_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取每個完整座標的絕對值，負值不直接加入或當作最大值；保留原向量符號。",
        "en": "Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector."
      },
      "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": "induced distance:all coordinate absolute values=[10, 1, 3]",
      "after": "maximum=10;attained coordinate=0",
      "relation": "implies",
      "rule_id": "linfinity_complete_maximum_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非空有限座標中最大值大於等於全部值且由某一座標達到；不是取最大帶號座標。",
        "en": "In a nonempty finite list,the maximum bounds every value and is attained at a listed coordinate.It is not the largest signed coordinate."
      },
      "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": [
      "linfinity",
      [
        5,
        0,
        2
      ],
      [
        -5,
        1,
        -1
      ],
      [
        10,
        -1,
        3
      ],
      10
    ]
  },
  "conclusion": {
    "canonical_value": "(linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 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-20261013-advance-20261026-r02-real-normed-vector-space-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations/questions.jsonl",
      "batch_sha256": "d235a859d0144feeea41a34962f61889794a71a7b28bff1d41a624d35aeabd92",
      "original_line": 93,
      "original_line_sha256": "4d662a4cfcb398b05da48a1be08d4d20ac04b22a04f5db78ef8fe96a78293987",
      "question_compact_sha256": "40960d8b5bb350e6f5bac09a732ecbd1cb876a5b3f7d8bc014b87461a4541bad",
      "english_row_compact_sha256": "1a6cca01e5d2b5a4349cc7f9831da44a554d1f1cd042c82c69ab0414320c71f3",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-real-normed-vector-space": {
        "sha256": "8f2547c8a3312254b34134e96a3083f313e6e7334ab0a87a1ba73cc147931964",
        "english_sha256": "2d1073f7489e9288fffcc9421a16d5080d3a63d7d3ae3f185810c959313006e9",
        "original_path": "public/data/mko/real-normed-vector-space.json",
        "english_path": "public/data/i18n/editions/mko-real-normed-vector-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "17a80c8482f64498e5a88bd4368d0a11b92d45c259784a0a536a29fbb98662d5",
    "original_candidate_sha256": "6742a47800f054bfef591a2fe0f6ab54cdc708243ca5b741a238499bba8bb66a",
    "original_candidate_line_sha256": "6742a47800f054bfef591a2fe0f6ab54cdc708243ca5b741a238499bba8bb66a"
  },
  "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-20261026-006-p25",
    "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": "1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35",
      "question_first_witness_sha256": "05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "a6ac836f34edd913ecd5a27432dc870672354b11a4f6b1f841d3e2ff6747a7ce",
      "arithmetic_steps": 3
    },
    "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."
  }
}