{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261027-r02-norm-induced-metric-0024",
  "pilot_number": "R0025-13124",
  "number": 13124,
  "family_id": "qf-mod27-normmetric-distance",
  "title": {
    "zh-Hant": "第 13124 題：範數誘導度量",
    "en": "Question 13124: The Metric Induced by a Fixed Named Norm"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261027-r02-norm-induced-metric-0024",
    "batch_id": "batch-20261014-advance-20261027-r02-norm-induced-metric-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-norm-induced-metric"
    ],
    "title_zh": "範數誘導度量",
    "stem_zh": "同一已認證範數以d(x,y)=norm(x-y)定義距離。操作=normmetric-distance；資料={\"norm_name\":\"linfinity\",\"left\":[11,-1,-2],\"right\":[-11,5,1]}。請依 [norm_name,left,right,difference,distance] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)",
      "explanation_zh": "Wave4根節點核對=(linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；誘導性來自範數公理的全稱證書；不宣稱每個metric皆由norm誘導或空間完備。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "範數誘導度量"
      ]
    },
    "variant": {
      "family_id": "qf-mod27-normmetric-distance",
      "seed": 202610272024,
      "parameters": {
        "norm_name": "linfinity",
        "left": [
          11,
          -1,
          -2
        ],
        "right": [
          -11,
          5,
          1
        ],
        "domain": "Named_norm_induced_metric_on_Rn",
        "operation": "normmetric-distance",
        "semantic_template": "qf-mod27-normmetric-distance/{\"left\":[11,-1,-2],\"norm_name\":\"linfinity\",\"right\":[-11,5,1]}"
      }
    },
    "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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r02-norm-induced-metric-0024",
    "batch_id": "batch-20261014-advance-20261027-r02-norm-induced-metric-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "cf81737cded244f1ae99dc3dd26f46d3029777091e9fbdb9fc270b353682f9af",
      "family_id": "qf-mod27-normmetric-distance",
      "seed": 202610272024,
      "semantic_template": "qf-mod27-normmetric-distance/{\"left\":[11,-1,-2],\"norm_name\":\"linfinity\",\"right\":[-11,5,1]}"
    },
    "fields": {
      "title": "The Metric Induced by a Fixed Named Norm",
      "stem": "Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[11, -1, -2], right=[-11, 5, 1]. Give (norm_name,left,right,left−right,distance).",
      "answer": {
        "kind": "expression",
        "value": "(linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)",
        "explanation": "Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)."
      },
      "skills": [
        "The Metric Induced by a Fixed Named Norm"
      ],
      "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": "同一已認證範數以d(x,y)=norm(x-y)定義距離。操作=normmetric-distance；資料={\"norm_name\":\"linfinity\",\"left\":[11,-1,-2],\"right\":[-11,5,1]}。請依 [norm_name,left,right,difference,distance] 欄序回答。",
        "en": "Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[11, -1, -2], right=[-11, 5, 1]. Give (norm_name,left,right,left−right,distance)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "requested distance difference coordinate0 := sub(11, -11)",
      "after": "requested distance difference coordinate0 = 22",
      "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 requested distance difference coordinate0，不以小數近似替代。",
          "en": "Recompute requested distance difference coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "-11"
        ],
        "output": "22",
        "label": "requested distance difference coordinate0"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "requested distance difference coordinate1 := sub(-1, 5)",
      "after": "requested distance difference coordinate1 = -6",
      "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 requested distance difference coordinate1，不以小數近似替代。",
          "en": "Recompute requested distance difference coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "5"
        ],
        "output": "-6",
        "label": "requested distance difference coordinate1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "requested distance difference coordinate2 := sub(-2, 1)",
      "after": "requested 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 requested distance difference coordinate2，不以小數近似替代。",
          "en": "Recompute requested distance difference coordinate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-3",
        "label": "requested distance difference coordinate2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "requested distance:norm=linfinity;left=[11, -1, -2];right=[-11, 5, 1]",
      "after": "signed left-right=[22, -6, -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": "requested distance:coordinate=0;signed value=22",
      "after": "absolute value=22",
      "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": "requested distance:coordinate=1;signed value=-6",
      "after": "absolute value=6",
      "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": "requested 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": "requested distance:all absolute coordinates=[22, 6, 3]",
      "after": "maximum=22;first attaining coordinate=0",
      "relation": "implies",
      "rule_id": "linfinity_full_maximum_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "最大值需大於等於所有項且確由一項達到；非空有限座標保證這一點。",
        "en": "The maximum must bound every entry and be attained at one entry,as guaranteed for nonempty finite coordinates."
      },
      "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",
      [
        11,
        -1,
        -2
      ],
      [
        -11,
        5,
        1
      ],
      [
        22,
        -6,
        -3
      ],
      22
    ]
  },
  "conclusion": {
    "canonical_value": "(linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)",
    "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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    "question_binding": {
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      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261027-r02-norm-induced-metric-foundations/questions.jsonl",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261027-r02-norm-induced-metric-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261027-r02-norm-induced-metric-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-norm-induced-metric": {
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        "english_sha256": "b7e34d77c8ec2850d00096ccb3e031e9c6b20a860974880e45e7fe95d8bbb5bb",
        "original_path": "public/data/mko/norm-induced-metric.json",
        "english_path": "public/data/i18n/editions/mko-norm-induced-metric.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a02aa2dccf790869cf864b821d0e60119686214561153daba232d921e07e7924",
    "original_candidate_sha256": "96e3bf919fc8bf2e4967f96a05852701723b90b416e0d6a2168f1430f777b5de",
    "original_candidate_line_sha256": "96e3bf919fc8bf2e4967f96a05852701723b90b416e0d6a2168f1430f777b5de"
  },
  "historical_source_state": {
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      "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-20261026-007-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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      "question_first_witness_sha256": "ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a"
    },
    "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": "b845126d90ea4752e675d853deadebc4f5a30e438d8cb2d5f9f1f0181668ca3c",
      "arithmetic_steps": 3
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    "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."
  }
}