{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261028-r05-bounded-linear-operator-0025",
  "pilot_number": "R0026-13975",
  "number": 13975,
  "family_id": "qf-mod28-operator-l1-norm",
  "title": {
    "zh-Hant": "第 13975 題：有界線性算子",
    "en": "Question 13975: The Exact l1 Operator Norm Uses Absolute Column Sums"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261028-r05-bounded-linear-operator-0025",
    "batch_id": "batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-bounded-linear-operator"
    ],
    "title_zh": "有界線性算子",
    "stem_zh": "來源與目標norm明示且使用單一統一常數。操作=operator-l1-norm；資料={\"matrix\":[[12,0,1],[2,-12,-3]]}。請依 [matrix,column_sums,operator_norm,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum)",
      "explanation_zh": "Wave4第二層核對=([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum)；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；矩陣誘導operator norm給全稱界證書；instance通過或非最佳上界不冒充exact norm。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有界線性算子"
      ]
    },
    "variant": {
      "family_id": "qf-mod28-operator-l1-norm",
      "seed": 202610285025,
      "parameters": {
        "matrix": [
          [
            12,
            0,
            1
          ],
          [
            2,
            -12,
            -3
          ]
        ],
        "domain": "Exact_matrix_induced_operator_norm",
        "operation": "operator-l1-norm",
        "semantic_template": "qf-mod28-operator-l1-norm/{\"matrix\":[[12,0,1],[2,-12,-3]]}"
      }
    },
    "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-r05-bounded-linear-operator-0025",
    "batch_id": "batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "30fa2607beec85ee2fab3cac5734593137cc2ee3a5d0ddbc277f2bd6e2d6d0b8",
      "family_id": "qf-mod28-operator-l1-norm",
      "seed": 202610285025,
      "semantic_template": "qf-mod28-operator-l1-norm/{\"matrix\":[[12,0,1],[2,-12,-3]]}"
    },
    "fields": {
      "title": "The Exact l1 Operator Norm Uses Absolute Column Sums",
      "stem": "Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.",
      "answer": {
        "kind": "expression",
        "value": "([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum)",
        "explanation": "Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum)."
      },
      "skills": [
        "The Exact l1 Operator Norm Uses Absolute Column Sums"
      ],
      "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": "來源與目標norm明示且使用單一統一常數。操作=operator-l1-norm；資料={\"matrix\":[[12,0,1],[2,-12,-3]]}。請依 [matrix,column_sums,operator_norm,certificate] 欄序回答。",
        "en": "Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "original column index=0;ordered entries=[12, 2]",
      "after": "absolute entries=[12, 2]",
      "relation": "implies",
      "rule_id": "operator_correct_axis_absolute_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "l1沿每欄往下加，linfinity沿每列橫加；保留原軸次序，不因長方形而混交換行列。",
        "en": "For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles."
      },
      "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": "operator column0 absolute sum := sum(12, 2)",
      "after": "operator column0 absolute sum = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 operator column0 absolute sum，不以小數近似替代。",
          "en": "Recompute operator column0 absolute sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "2"
        ],
        "output": "14",
        "label": "operator column0 absolute sum"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original column index=1;ordered entries=[0, -12]",
      "after": "absolute entries=[0, 12]",
      "relation": "implies",
      "rule_id": "operator_correct_axis_absolute_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "l1沿每欄往下加，linfinity沿每列橫加；保留原軸次序，不因長方形而混交換行列。",
        "en": "For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "operator column1 absolute sum := sum(0, 12)",
      "after": "operator column1 absolute sum = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 operator column1 absolute sum，不以小數近似替代。",
          "en": "Recompute operator column1 absolute sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "12"
        ],
        "output": "12",
        "label": "operator column1 absolute sum"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original column index=2;ordered entries=[1, -3]",
      "after": "absolute entries=[1, 3]",
      "relation": "implies",
      "rule_id": "operator_correct_axis_absolute_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "l1沿每欄往下加，linfinity沿每列橫加；保留原軸次序，不因長方形而混交換行列。",
        "en": "For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles."
      },
      "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": "operator column2 absolute sum := sum(1, 3)",
      "after": "operator column2 absolute sum = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 operator column2 absolute sum，不以小數近似替代。",
          "en": "Recompute operator column2 absolute sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "3"
        ],
        "output": "4",
        "label": "operator column2 absolute sum"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "all absolute column sums=[14, 12, 4]",
      "after": "maximum=14;first maximizing column=0",
      "relation": "implies",
      "rule_id": "operator_maximum_axis_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算完所有指定軸的和，才取最大值；這個值稍後要同時有全稱上界與達到的單位輸入。",
        "en": "Compute every required axis sum before taking a maximum.The value must have both a universal upper bound and an attaining unit 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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "arbitrary x in R^3;all column sums<=14",
      "after": "norm1(Ax)<=sum_j(sum_i abs(Aij))*abs(xj)<=14*norm1(x)",
      "relation": "implies",
      "rule_id": "operator_l1_universal_column_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐輸出用三角不等式，再交換有限雙重和，各欄係數≤最大欄和L，故對每個實向量都有同一常數界，不是抽樣。",
        "en": "Apply the triangle inequality to each output,exchange finite sums,and bound every column coefficient by L.This yields one constant for every real input,not samples."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "attaining unit input image:matrix=[[12, 0, 1], [2, -12, -3]];input=[1, 0, 0]",
      "after": "input dimension=3;output dimension=2;rows dot input",
      "relation": "implies",
      "rule_id": "matrix_row_column_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留長方形矩陣的欄輸入、列輸出角色，計算值可超過原輸入座標界，不重新截斷。",
        "en": "Retain column-input/row-output roles for rectangular matrices.Computed values can exceed raw input bounds and are not clipped."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "attaining unit input image row0 column0 product := mul(12, 1)",
      "after": "attaining unit input image row0 column0 product = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining unit input image row0 column0 product，不以小數近似替代。",
          "en": "Recompute attaining unit input image row0 column0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "1"
        ],
        "output": "12",
        "label": "attaining unit input image row0 column0 product"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "attaining unit input image row0 column1 product := mul(0, 0)",
      "after": "attaining unit input image row0 column1 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining unit input image row0 column1 product，不以小數近似替代。",
          "en": "Recompute attaining unit input image row0 column1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "attaining unit input image row0 column1 product"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "attaining unit input image row0 column2 product := mul(1, 0)",
      "after": "attaining unit input image row0 column2 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining unit input image row0 column2 product，不以小數近似替代。",
          "en": "Recompute attaining unit input image row0 column2 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "attaining unit input image row0 column2 product"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "attaining unit input image row0 sum := sum(12, 0, 0)",
      "after": "attaining unit input image row0 sum = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining unit input image row0 sum，不以小數近似替代。",
          "en": "Recompute attaining unit input image row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "0",
          "0"
        ],
        "output": "12",
        "label": "attaining unit input image row0 sum"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "attaining unit input image row1 column0 product := mul(2, 1)",
      "after": "attaining unit input image row1 column0 product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining unit input image row1 column0 product，不以小數近似替代。",
          "en": "Recompute attaining unit input image row1 column0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "attaining unit input image row1 column0 product"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "attaining unit input image row1 column1 product := mul(-12, 0)",
      "after": "attaining unit input image row1 column1 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining unit input image row1 column1 product，不以小數近似替代。",
          "en": "Recompute attaining unit input image row1 column1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-12",
          "0"
        ],
        "output": "0",
        "label": "attaining unit input image row1 column1 product"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "attaining unit input image row1 column2 product := mul(-3, 0)",
      "after": "attaining unit input image row1 column2 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining unit input image row1 column2 product，不以小數近似替代。",
          "en": "Recompute attaining unit input image row1 column2 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "0"
        ],
        "output": "0",
        "label": "attaining unit input image row1 column2 product"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "attaining unit input image row1 sum := sum(2, 0, 0)",
      "after": "attaining unit input image row1 sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining unit input image row1 sum，不以小數近似替代。",
          "en": "Recompute attaining unit input image row1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "0",
          "0"
        ],
        "output": "2",
        "label": "attaining unit input image row1 sum"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "attaining input:signed vector=[1, 0, 0];norm=l1",
      "after": "coordinate absolute values=[1, 0, 0]",
      "relation": "implies",
      "rule_id": "operator_named_norm_original_signs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先對完整座標取絕對值，不改原向量配對；這裡也可接收已算出的矩陣像，而不套原輸入界截斷。",
        "en": "Take absolute values of whole coordinates without changing their pairing.This also accepts computed matrix images without clipping to raw input bounds."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "attaining input l1 sum := sum(1, 0, 0)",
      "after": "attaining input l1 sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining input l1 sum，不以小數近似替代。",
          "en": "Recompute attaining input l1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "0",
          "0"
        ],
        "output": "1",
        "label": "attaining input l1 sum"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "attaining output:signed vector=[12, 2];norm=l1",
      "after": "coordinate absolute values=[12, 2]",
      "relation": "implies",
      "rule_id": "operator_named_norm_original_signs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先對完整座標取絕對值，不改原向量配對；這裡也可接收已算出的矩陣像，而不套原輸入界截斷。",
        "en": "Take absolute values of whole coordinates without changing their pairing.This also accepts computed matrix images without clipping to raw input bounds."
      },
      "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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "attaining output l1 sum := sum(12, 2)",
      "after": "attaining output l1 sum = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 attaining output l1 sum，不以小數近似替代。",
          "en": "Recompute attaining output l1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "2"
        ],
        "output": "14",
        "label": "attaining output l1 sum"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "explicit input=[1, 0, 0];input norm=1;image=[12, 2];output norm=14",
      "after": "uniform upper bound=14 and attained ratio=14;exact induced norm=14",
      "relation": "implies",
      "rule_id": "operator_attainment_proves_least_uniform_constant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "l1取最大欄的座標基，linfinity取最大列的係數符號；零矩陣仍選非零單位向量。已證所有輸入上界L，又有單位輸入達L，所以更小常數不可能，才是exact norm。",
        "en": "For l1 take the maximizing column's basis vector;for linfinity take signs of a maximizing row.For a zero matrix still choose a nonzero unit vector.The universal bound L and an attained unit ratio L exclude every smaller constant,establishing the exact norm."
      },
      "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": [
      [
        [
          12,
          0,
          1
        ],
        [
          2,
          -12,
          -3
        ]
      ],
      [
        14,
        12,
        4
      ],
      14,
      "maximum_absolute_column_sum"
    ]
  },
  "conclusion": {
    "canonical_value": "([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum)",
    "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-20261014-advance-20261028-r05-bounded-linear-operator-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations/questions.jsonl",
      "batch_sha256": "4e3983c67a61725eb4b26fcdd27afef70ba89f3bf5f2e573303953552aa0d83d",
      "original_line": 25,
      "original_line_sha256": "b832534278d022e9fcf9fad5c1f6f9a895f4308e3c6e8a5af7c29d4863b95ef3",
      "question_compact_sha256": "30fa2607beec85ee2fab3cac5734593137cc2ee3a5d0ddbc277f2bd6e2d6d0b8",
      "english_row_compact_sha256": "9b26d9fe4c89c31ede76689727fdeac1d1b67e112ff28fafb2e9d1855a182fed",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-bounded-linear-operator": {
        "sha256": "c673fba2cd8ab7e707c55dc85252e9d84fa9bed89d6cfd51ac95bcc4cee707f9",
        "english_sha256": "3ac597900945ab4e80b91b57863d6afe3de538143013f1cd21732bdc4e4906a2",
        "original_path": "public/data/mko/bounded-linear-operator.json",
        "english_path": "public/data/i18n/editions/mko-bounded-linear-operator.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e9c0ae2888b9f0f177728712cf7eb267a13a2a12f233a4c64ea50cb104f58bb0",
    "original_candidate_sha256": "7d0f67a7e7dd2c4516f64dd41024a8299fd4ce79988c5f901f852d2f57274fee",
    "original_candidate_line_sha256": "7d0f67a7e7dd2c4516f64dd41024a8299fd4ce79988c5f901f852d2f57274fee"
  },
  "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-20261026-007-p27",
    "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": "f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798",
      "question_first_witness_sha256": "cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "9e62b6b4380f259ed009b51ca969dd23662ab19a0b75ebfded30afd0ea803b94",
      "arithmetic_steps": 13
    },
    "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."
  }
}