{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261105-r01-direct-sum-0049",
  "pilot_number": "R0036-18624",
  "number": 18624,
  "family_id": "qf-mod36-directsum-ambient-decomposition",
  "title": {
    "zh-Hant": "第 18624 題：子空間直和",
    "en": "Question 18624: Check This Decomposition Separately from a Whole-Space Direct Sum"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261105-r01-direct-sum-0049",
    "batch_id": "batch-20261014-advance-20261105-r01-direct-sum-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-direct-sum"
    ],
    "title_zh": "子空間直和",
    "stem_zh": "有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,24]],\"vector\":[24,2,48],\"coefficients_u\":[24],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([24, 0, 0], [0, 2, 48], [24, 2, 48], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)",
      "explanation_zh": "Wave5第五層核對=([24, 0, 0], [0, 2, 48], [24, 2, 48], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "子空間直和"
      ]
    },
    "variant": {
      "family_id": "qf-mod36-directsum-ambient-decomposition",
      "seed": 202611051049,
      "parameters": {
        "ambient_dimension": 3,
        "basis_u": [
          [
            1,
            0,
            0
          ]
        ],
        "basis_w": [
          [
            0,
            1,
            24
          ]
        ],
        "vector": [
          24,
          2,
          48
        ],
        "coefficients_u": [
          24
        ],
        "coefficients_w": [
          2
        ],
        "domain": "Exact_finite_dimensional_real_subspaces",
        "operation": "directsum-ambient-decomposition",
        "semantic_template": "qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,24]],\"coefficients_u\":[24],\"coefficients_w\":[2],\"vector\":[24,2,48]}"
      }
    },
    "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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r01-direct-sum-0049",
    "batch_id": "batch-20261014-advance-20261105-r01-direct-sum-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "97407ed5167120113e37eead4a5c585cf77bf7a4b63bbd99a465a41f7fd3a39b",
      "family_id": "qf-mod36-directsum-ambient-decomposition",
      "seed": 202611051049,
      "semantic_template": "qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,24]],\"coefficients_u\":[24],\"coefficients_w\":[2],\"vector\":[24,2,48]}"
    },
    "fields": {
      "title": "Check This Decomposition Separately from a Whole-Space Direct Sum",
      "stem": "Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 24]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[24, 2, 48], coefficients_u=[24], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([24, 0, 0], [0, 2, 48], [24, 2, 48], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)",
        "explanation": "Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([24, 0, 0], [0, 2, 48], [24, 2, 48], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."
      },
      "skills": [
        "Check This Decomposition Separately from a Whole-Space Direct Sum"
      ],
      "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": "有限維實向量空間、完整基底與exact rank證書。操作=directsum-ambient-decomposition；資料={\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,24]],\"vector\":[24,2,48],\"coefficients_u\":[24],\"coefficients_w\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 24]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[24, 2, 48], coefficients_u=[24], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=directsum-ambient-decomposition;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。",
        "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence."
      },
      "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": "directsum U:complete ordered matrix=[[1], [0], [0]];rows=3;columns=1",
      "after": "use exact invertible ROW operations only;original COLUMN roles/order stay fixed",
      "relation": "implies",
      "rule_id": "finite_linear_source_matrix_not_semantic_reordering",
      "conditions": [],
      "explanation": {
        "zh-Hant": "基向量放欄且保原順序；semantic用的排序不能拿來重排係數或候選泛函。列運算是左乘可逆映射，保核、欄相依及rank，不宣稱實際欄空間字面不變。",
        "en": "Keep original basis vectors in ordered columns.Semantic sorting must not reorder coefficients or candidate rows.Row operations multiply by an invertible map,retaining kernel,column relations and rank,not the literal original column space."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "directsum U pivot0 column0 normalized entry0 := div(1, 1)",
      "after": "directsum U pivot0 column0 normalized entry0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum U pivot0 column0 normalized entry0，不以小數近似替代。",
          "en": "Recompute directsum U pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "directsum U pivot0 column0 normalized entry0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "directsum U:pivot row=0;column=0;nonzero pivot=1;old row=[1]",
      "after": "normalized full row=[1]",
      "relation": "implies",
      "rule_id": "finite_linear_nonzero_pivot_complete_row_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列每格同除非零pivot，精確有理數保零值與符號。",
        "en": "Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs."
      },
      "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": "directsum U:complete exact reduced matrix=[[1], [0], [0]];pivot columns=[0]",
      "after": "rank=1;pivot columns independent and all original column relations preserved;zero rows retained",
      "relation": "implies",
      "rule_id": "finite_linear_complete_rank_certificate_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "主元欄有不同的首1座標而獨立，其餘欄由主元列座標生成；可逆列運算把依賴關係帶回原矩陣。零列不刪除，不從有限向量測試猜rank。",
        "en": "Distinct pivot identity coordinates prove independence;other columns are generated by their pivot-row coordinates.Invertible operations transport column relations back to the original matrix.Keep zero rows;do not infer rank from sampled vectors."
      },
      "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": "directsum U:complete ORIGINAL independent columns=[[1, 0, 0]]",
      "after": "declared subspace dimension=1;basis of its full real span;empty basis means zero subspace",
      "relation": "implies",
      "rule_id": "directsum_declared_basis_complete_real_span",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原資料宣告完整basis，精確rank再核獨立，故數量才是此子空間維度；允許空basis代表零子空間，不把有限列表當子空間所有元素。",
        "en": "The source declares a full basis and exact rank checks independence,so its count is the subspace dimension.An empty basis spans the zero subspace;the finite list is not the set of all subspace elements."
      },
      "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": "directsum W:complete ordered matrix=[[0], [1], [24]];rows=3;columns=1",
      "after": "use exact invertible ROW operations only;original COLUMN roles/order stay fixed",
      "relation": "implies",
      "rule_id": "finite_linear_source_matrix_not_semantic_reordering",
      "conditions": [],
      "explanation": {
        "zh-Hant": "基向量放欄且保原順序；semantic用的排序不能拿來重排係數或候選泛函。列運算是左乘可逆映射，保核、欄相依及rank，不宣稱實際欄空間字面不變。",
        "en": "Keep original basis vectors in ordered columns.Semantic sorting must not reorder coefficients or candidate rows.Row operations multiply by an invertible map,retaining kernel,column relations and rank,not the literal original column space."
      },
      "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": "directsum W:matrix=[[0], [1], [24]];selected row=1",
      "after": "swap rows0,1;matrix=[[1], [0], [24]]",
      "relation": "implies",
      "rule_id": "finite_linear_invertible_row_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列交換可逆，原欄與其係數標籤不變。",
        "en": "Swap entire rows invertibly while retaining original column/coefficient labels."
      },
      "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": "directsum W pivot0 column0 normalized entry0 := div(1, 1)",
      "after": "directsum W pivot0 column0 normalized entry0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum W pivot0 column0 normalized entry0，不以小數近似替代。",
          "en": "Recompute directsum W pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "directsum W pivot0 column0 normalized entry0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "directsum W:pivot row=0;column=0;nonzero pivot=1;old row=[1]",
      "after": "normalized full row=[1]",
      "relation": "implies",
      "rule_id": "finite_linear_nonzero_pivot_complete_row_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列每格同除非零pivot，精確有理數保零值與符號。",
        "en": "Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "directsum W eliminate row2 pivot0 entry0 product := mul(24, 1)",
      "after": "directsum W eliminate row2 pivot0 entry0 product = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum W eliminate row2 pivot0 entry0 product，不以小數近似替代。",
          "en": "Recompute directsum W eliminate row2 pivot0 entry0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "1"
        ],
        "output": "24",
        "label": "directsum W eliminate row2 pivot0 entry0 product"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "directsum W eliminate row2 pivot0 entry0 difference := sub(24, 24)",
      "after": "directsum W eliminate row2 pivot0 entry0 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum W eliminate row2 pivot0 entry0 difference，不以小數近似替代。",
          "en": "Recompute directsum W eliminate row2 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "24",
          "24"
        ],
        "output": "0",
        "label": "directsum W eliminate row2 pivot0 entry0 difference"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "directsum W:target row=2;old=[24];factor=24;pivot row=[1]",
      "after": "new full row=[0]",
      "relation": "implies",
      "rule_id": "finite_linear_invertible_all_entry_elimination",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一列的所有格都減同一倍pivot列；反向加回即逆運算。",
        "en": "Subtract the same pivot-row multiple in every entry;adding it back is the inverse."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "directsum W:complete exact reduced matrix=[[1], [0], [0]];pivot columns=[0]",
      "after": "rank=1;pivot columns independent and all original column relations preserved;zero rows retained",
      "relation": "implies",
      "rule_id": "finite_linear_complete_rank_certificate_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "主元欄有不同的首1座標而獨立，其餘欄由主元列座標生成；可逆列運算把依賴關係帶回原矩陣。零列不刪除，不從有限向量測試猜rank。",
        "en": "Distinct pivot identity coordinates prove independence;other columns are generated by their pivot-row coordinates.Invertible operations transport column relations back to the original matrix.Keep zero rows;do not infer rank from sampled vectors."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "directsum W:complete ORIGINAL independent columns=[[0, 1, 24]]",
      "after": "declared subspace dimension=1;basis of its full real span;empty basis means zero subspace",
      "relation": "implies",
      "rule_id": "directsum_declared_basis_complete_real_span",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原資料宣告完整basis，精確rank再核獨立，故數量才是此子空間維度；允許空basis代表零子空間，不把有限列表當子空間所有元素。",
        "en": "The source declares a full basis and exact rank checks independence,so its count is the subspace dimension.An empty basis spans the zero subspace;the finite list is not the set of all subspace elements."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "directsum ordered U then W:complete ordered matrix=[[1, 0], [0, 1], [0, 24]];rows=3;columns=2",
      "after": "use exact invertible ROW operations only;original COLUMN roles/order stay fixed",
      "relation": "implies",
      "rule_id": "finite_linear_source_matrix_not_semantic_reordering",
      "conditions": [],
      "explanation": {
        "zh-Hant": "基向量放欄且保原順序；semantic用的排序不能拿來重排係數或候選泛函。列運算是左乘可逆映射，保核、欄相依及rank，不宣稱實際欄空間字面不變。",
        "en": "Keep original basis vectors in ordered columns.Semantic sorting must not reorder coefficients or candidate rows.Row operations multiply by an invertible map,retaining kernel,column relations and rank,not the literal original column space."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "directsum ordered U then W pivot0 column0 normalized entry0 := div(1, 1)",
      "after": "directsum ordered U then W pivot0 column0 normalized entry0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum ordered U then W pivot0 column0 normalized entry0，不以小數近似替代。",
          "en": "Recompute directsum ordered U then W pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "directsum ordered U then W pivot0 column0 normalized entry0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "directsum ordered U then W pivot0 column0 normalized entry1 := div(0, 1)",
      "after": "directsum ordered U then W pivot0 column0 normalized entry1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum ordered U then W pivot0 column0 normalized entry1，不以小數近似替代。",
          "en": "Recompute directsum ordered U then W pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "directsum ordered U then W pivot0 column0 normalized entry1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "directsum ordered U then W:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0]",
      "after": "normalized full row=[1, 0]",
      "relation": "implies",
      "rule_id": "finite_linear_nonzero_pivot_complete_row_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列每格同除非零pivot，精確有理數保零值與符號。",
        "en": "Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "directsum ordered U then W pivot1 column1 normalized entry0 := div(0, 1)",
      "after": "directsum ordered U then W pivot1 column1 normalized entry0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum ordered U then W pivot1 column1 normalized entry0，不以小數近似替代。",
          "en": "Recompute directsum ordered U then W pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "directsum ordered U then W pivot1 column1 normalized entry0"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "directsum ordered U then W pivot1 column1 normalized entry1 := div(1, 1)",
      "after": "directsum ordered U then W pivot1 column1 normalized entry1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum ordered U then W pivot1 column1 normalized entry1，不以小數近似替代。",
          "en": "Recompute directsum ordered U then W pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "directsum ordered U then W pivot1 column1 normalized entry1"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "directsum ordered U then W:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1]",
      "after": "normalized full row=[0, 1]",
      "relation": "implies",
      "rule_id": "finite_linear_nonzero_pivot_complete_row_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列每格同除非零pivot，精確有理數保零值與符號。",
        "en": "Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs."
      },
      "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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "directsum ordered U then W eliminate row2 pivot1 entry0 product := mul(24, 0)",
      "after": "directsum ordered U then W eliminate row2 pivot1 entry0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum ordered U then W eliminate row2 pivot1 entry0 product，不以小數近似替代。",
          "en": "Recompute directsum ordered U then W eliminate row2 pivot1 entry0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "0"
        ],
        "output": "0",
        "label": "directsum ordered U then W eliminate row2 pivot1 entry0 product"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "directsum ordered U then W eliminate row2 pivot1 entry0 difference := sub(0, 0)",
      "after": "directsum ordered U then W eliminate row2 pivot1 entry0 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum ordered U then W eliminate row2 pivot1 entry0 difference，不以小數近似替代。",
          "en": "Recompute directsum ordered U then W eliminate row2 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "directsum ordered U then W eliminate row2 pivot1 entry0 difference"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "directsum ordered U then W eliminate row2 pivot1 entry1 product := mul(24, 1)",
      "after": "directsum ordered U then W eliminate row2 pivot1 entry1 product = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum ordered U then W eliminate row2 pivot1 entry1 product，不以小數近似替代。",
          "en": "Recompute directsum ordered U then W eliminate row2 pivot1 entry1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "1"
        ],
        "output": "24",
        "label": "directsum ordered U then W eliminate row2 pivot1 entry1 product"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "directsum ordered U then W eliminate row2 pivot1 entry1 difference := sub(24, 24)",
      "after": "directsum ordered U then W eliminate row2 pivot1 entry1 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum ordered U then W eliminate row2 pivot1 entry1 difference，不以小數近似替代。",
          "en": "Recompute directsum ordered U then W eliminate row2 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "24",
          "24"
        ],
        "output": "0",
        "label": "directsum ordered U then W eliminate row2 pivot1 entry1 difference"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "directsum ordered U then W:target row=2;old=[0, 24];factor=24;pivot row=[0, 1]",
      "after": "new full row=[0, 0]",
      "relation": "implies",
      "rule_id": "finite_linear_invertible_all_entry_elimination",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一列的所有格都減同一倍pivot列；反向加回即逆運算。",
        "en": "Subtract the same pivot-row multiple in every entry;adding it back is the inverse."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "directsum ordered U then W:complete exact reduced matrix=[[1, 0], [0, 1], [0, 0]];pivot columns=[0, 1]",
      "after": "rank=2;pivot columns independent and all original column relations preserved;zero rows retained",
      "relation": "implies",
      "rule_id": "finite_linear_complete_rank_certificate_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "主元欄有不同的首1座標而獨立，其餘欄由主元列座標生成；可逆列運算把依賴關係帶回原矩陣。零列不刪除，不從有限向量測試猜rank。",
        "en": "Distinct pivot identity coordinates prove independence;other columns are generated by their pivot-row coordinates.Invertible operations transport column relations back to the original matrix.Keep zero rows;do not infer rank from sampled vectors."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "directsum U plus W:pivot columns=[0, 1];ALL free columns=[];full relation family=[]",
      "after": "EVERY zero combination is uniquely determined by its free coordinates;these free-coordinate unit relations form the full kernel;dimension=0",
      "relation": "implies",
      "rule_id": "directsum_full_relation_space_both_directions",
      "conditions": [],
      "explanation": {
        "zh-Hant": "可逆列運算保原矩陣核，任意零組合的自由座標就是此族的組合係數，主元方程定其餘座標；反向代回皆零。自由座標identity亦證獨立，故不只列幾個樣本關係。",
        "en": "Invertible row operations preserve the original kernel.Every zero combination's free coordinates are its unique coefficients in this family,and pivot equations determine the rest.Conversely all these combinations map to zero.The free-coordinate identity proves independence,not merely sampled relations."
      },
      "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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "directsum U and W basis count sum := sum(1, 1)",
      "after": "directsum U and W basis count sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum U and W basis count sum，不以小數近似替代。",
          "en": "Recompute directsum U and W basis count sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "directsum U and W basis count sum"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "directsum actual intersection dimension := sub(2, 2)",
      "after": "directsum actual intersection dimension = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum actual intersection dimension，不以小數近似替代。",
          "en": "Recompute directsum actual intersection dimension with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "directsum actual intersection dimension"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "ORIGINAL U basis=[[1, 0, 0]];W basis=[[0, 1, 24]];combined rank=2;full zero-combination relations=[];common vectors=[]",
      "after": "kernel of sum map consists of(u,-u) for u in U intersection W;intersection dimension=0;unique decomposition INSIDE U+W=是;coverage of R^3=否",
      "relation": "implies",
      "rule_id": "directsum_unique_in_sum_and_ambient_coverage_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "和映射滿射U+W，核與U∩W經v↦(v,−v)互逆線性對應；完整核basis因此給交集維度。分解差在此核，所以交集0才是和內唯一。只有再加rank=ambient dimension才是整個母空間直和，不能由一個target碰巧重建成功外推。",
        "en": "The sum map surjects onto U+W,and its kernel is linearly isomorphic to U∩W via v↦(v,−v).The complete kernel basis gives the intersection dimension.Decomposition differences lie in this kernel,so zero intersection gives uniqueness within the sum.Full ambient directness additionally requires rank equal to ambient dimension,not one successfully reconstructed target."
      },
      "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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "directsum target U coordinate0 coordinate0 product := mul(1, 24)",
      "after": "directsum target U coordinate0 coordinate0 product = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target U coordinate0 coordinate0 product，不以小數近似替代。",
          "en": "Recompute directsum target U coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "24"
        ],
        "output": "24",
        "label": "directsum target U coordinate0 coordinate0 product"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "directsum target U coordinate0 complete sum := sum(24)",
      "after": "directsum target U coordinate0 complete sum = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target U coordinate0 complete sum，不以小數近似替代。",
          "en": "Recompute directsum target U coordinate0 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24"
        ],
        "output": "24",
        "label": "directsum target U coordinate0 complete sum"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "directsum target U coordinate1 coordinate0 product := mul(0, 24)",
      "after": "directsum target U coordinate1 coordinate0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target U coordinate1 coordinate0 product，不以小數近似替代。",
          "en": "Recompute directsum target U coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "24"
        ],
        "output": "0",
        "label": "directsum target U coordinate1 coordinate0 product"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "directsum target U coordinate1 complete sum := sum(0)",
      "after": "directsum target U coordinate1 complete sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target U coordinate1 complete sum，不以小數近似替代。",
          "en": "Recompute directsum target U coordinate1 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "directsum target U coordinate1 complete sum"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "directsum target U coordinate2 coordinate0 product := mul(0, 24)",
      "after": "directsum target U coordinate2 coordinate0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target U coordinate2 coordinate0 product，不以小數近似替代。",
          "en": "Recompute directsum target U coordinate2 coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "24"
        ],
        "output": "0",
        "label": "directsum target U coordinate2 coordinate0 product"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "directsum target U coordinate2 complete sum := sum(0)",
      "after": "directsum target U coordinate2 complete sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target U coordinate2 complete sum，不以小數近似替代。",
          "en": "Recompute directsum target U coordinate2 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "directsum target U coordinate2 complete sum"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "directsum target W coordinate0 coordinate0 product := mul(0, 2)",
      "after": "directsum target W coordinate0 coordinate0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target W coordinate0 coordinate0 product，不以小數近似替代。",
          "en": "Recompute directsum target W coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "directsum target W coordinate0 coordinate0 product"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "directsum target W coordinate0 complete sum := sum(0)",
      "after": "directsum target W coordinate0 complete sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target W coordinate0 complete sum，不以小數近似替代。",
          "en": "Recompute directsum target W coordinate0 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "directsum target W coordinate0 complete sum"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "directsum target W coordinate1 coordinate0 product := mul(1, 2)",
      "after": "directsum target W coordinate1 coordinate0 product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target W coordinate1 coordinate0 product，不以小數近似替代。",
          "en": "Recompute directsum target W coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "directsum target W coordinate1 coordinate0 product"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "directsum target W coordinate1 complete sum := sum(2)",
      "after": "directsum target W coordinate1 complete sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target W coordinate1 complete sum，不以小數近似替代。",
          "en": "Recompute directsum target W coordinate1 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2"
        ],
        "output": "2",
        "label": "directsum target W coordinate1 complete sum"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "directsum target W coordinate2 coordinate0 product := mul(24, 2)",
      "after": "directsum target W coordinate2 coordinate0 product = 48",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target W coordinate2 coordinate0 product，不以小數近似替代。",
          "en": "Recompute directsum target W coordinate2 coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "2"
        ],
        "output": "48",
        "label": "directsum target W coordinate2 coordinate0 product"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "directsum target W coordinate2 complete sum := sum(48)",
      "after": "directsum target W coordinate2 complete sum = 48",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target W coordinate2 complete sum，不以小數近似替代。",
          "en": "Recompute directsum target W coordinate2 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "48"
        ],
        "output": "48",
        "label": "directsum target W coordinate2 complete sum"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "directsum target reconstructed coordinate0 := sum(24, 0)",
      "after": "directsum target reconstructed coordinate0 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target reconstructed coordinate0，不以小數近似替代。",
          "en": "Recompute directsum target reconstructed coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "0"
        ],
        "output": "24",
        "label": "directsum target reconstructed coordinate0"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "directsum target reconstructed coordinate1 := sum(0, 2)",
      "after": "directsum target reconstructed coordinate1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target reconstructed coordinate1，不以小數近似替代。",
          "en": "Recompute directsum target reconstructed coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "directsum target reconstructed coordinate1"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "directsum target reconstructed coordinate2 := sum(0, 48)",
      "after": "directsum target reconstructed coordinate2 = 48",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 directsum target reconstructed coordinate2，不以小數近似替代。",
          "en": "Recompute directsum target reconstructed coordinate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "48"
        ],
        "output": "48",
        "label": "directsum target reconstructed coordinate2"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "original coefficient blocks U=[24],W=[2] in ORIGINAL basis order;target=[24, 2, 48]",
      "after": "computed parts=[[24, 0, 0], [0, 2, 48]];reconstructed=[24, 2, 48];matches THIS target=是;unique-in-sum=是;ambient coverage=否",
      "relation": "implies",
      "rule_id": "directsum_one_target_reconstruction_not_all_ambient_coverage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原basis順序做線性組合並逐座標比對原target；不以semantic排序重排係數，也不把一個相等當所有向量可分解。",
        "en": "Use the original basis order for both linear combinations and compare every coordinate with the original target.Do not reorder coefficients by semantic sorting or turn one equality into coverage of every 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
    }
  ],
  "math_model": {
    "result": [
      [
        "24",
        "0",
        "0"
      ],
      [
        "0",
        "2",
        "48"
      ],
      [
        "24",
        "2",
        "48"
      ],
      true,
      true,
      false,
      "whole_ambient_direct_sum_requires_uniqueness_and_coverage"
    ]
  },
  "conclusion": {
    "canonical_value": "([24, 0, 0], [0, 2, 48], [24, 2, 48], 是, 是, 否, whole_ambient_direct_sum_requires_uniqueness_and_coverage)",
    "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-20261105-r01-direct-sum-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r01-direct-sum-foundations/questions.jsonl",
      "batch_sha256": "7e0295c5944e58b0de639be2d989049e9f65b8f21905ff6597dbf939a5f3be6d",
      "original_line": 49,
      "original_line_sha256": "287410042f1ccfeb2e2766c5f5aa8c8d900f7202556f1649add5f2e13fc398ea",
      "question_compact_sha256": "97407ed5167120113e37eead4a5c585cf77bf7a4b63bbd99a465a41f7fd3a39b",
      "english_row_compact_sha256": "956d7cb59b0088b43cbd29ccff9998f377006e45a728fd802b3fe56a7d030696",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r01-direct-sum-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r01-direct-sum-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-direct-sum": {
        "sha256": "3eedbed2a646d34aa3bf6b0c5eb4e3482fcd8c816d40ca7c5652a6bf7224b9e8",
        "english_sha256": "1fe3402da619d87b341bcf2c97aad834c0b1ea87391581c56651a5dc46bc3779",
        "original_path": "public/data/mko/direct-sum.json",
        "english_path": "public/data/i18n/editions/mko-direct-sum.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994",
    "original_candidate_sha256": "e231ff988bafc71de0d46b79e1d149680d81cddc11b81a34a1759025837fe1ce",
    "original_candidate_line_sha256": "e231ff988bafc71de0d46b79e1d149680d81cddc11b81a34a1759025837fe1ce"
  },
  "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-20261026-008-p37",
    "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": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301",
      "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 48,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2262b6dc47784d6df77d3d2d0a6f022931c6ce5c52a8c83f58c6bf26e468c5f6",
      "arithmetic_steps": 29
    },
    "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."
  }
}