{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261029-r03-inner-product-induced-norm-0093",
  "pilot_number": "R0028-14568",
  "number": 14568,
  "family_id": "qf-mod29-innernorm-parallelogram",
  "title": {
    "zh-Hant": "第 14568 題：內積誘導範數",
    "en": "Question 14568: Check the Parallelogram Identity in One Inner Product"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261029-r03-inner-product-induced-norm-0093",
    "batch_id": "batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-inner-product-induced-norm"
    ],
    "title_zh": "內積誘導範數",
    "stem_zh": "向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}。請依 [x,y,lhs,rhs,equal] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 0], [1, -5], 248, 248, 是)",
      "explanation_zh": "Wave4第三層核對=([0, 0], [1, -5], 248, 248, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；範數為自內積非負平方根且三角責任依Cauchy–Schwarz；精確instances不替全稱證書且不推出完備。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "內積誘導範數"
      ]
    },
    "variant": {
      "family_id": "qf-mod29-innernorm-parallelogram",
      "seed": 202610293093,
      "parameters": {
        "matrix": [
          [
            19,
            2
          ],
          [
            2,
            5
          ]
        ],
        "x": [
          0,
          0
        ],
        "y": [
          1,
          -5
        ],
        "domain": "Exact_R2_inner_product_induced_norm",
        "operation": "innernorm-parallelogram",
        "semantic_template": "qf-mod29-innernorm-parallelogram/{\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r03-inner-product-induced-norm-0093",
    "batch_id": "batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2c13fc1e646bf7f13de812d5bfbee5d1528124c92f884854cce8be65ea7a9d84",
      "family_id": "qf-mod29-innernorm-parallelogram",
      "seed": 202610293093,
      "semantic_template": "qf-mod29-innernorm-parallelogram/{\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}"
    },
    "fields": {
      "title": "Check the Parallelogram Identity in One Inner Product",
      "stem": "Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[19, 2], [2, 5]] and x=[0, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[1, -5]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).",
      "answer": {
        "kind": "expression",
        "value": "([0, 0], [1, -5], 248, 248, Yes)",
        "explanation": "Evaluate all four self-inner-products in the same matrix. This identity is an inner-product consistency check, not a completeness or Hilbert certificate. The exact result, retaining the requested order, is ([0, 0], [1, -5], 248, 248, Yes)."
      },
      "skills": [
        "Check the Parallelogram Identity in One Inner Product"
      ],
      "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": "向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={\"matrix\":[[19,2],[2,5]],\"x\":[0,0],\"y\":[1,-5]}。請依 [x,y,lhs,rhs,equal] 欄序回答。",
        "en": "Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[19, 2], [2, 5]] and x=[0, 0]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[1, -5]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "inner parent diagonal product := mul(19, 5)",
      "after": "inner parent diagonal product = 95",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner parent diagonal product，不以小數近似替代。",
          "en": "Recompute inner parent diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "5"
        ],
        "output": "95",
        "label": "inner parent diagonal product"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "inner parent off-diagonal product := mul(2, 2)",
      "after": "inner parent off-diagonal product = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner parent off-diagonal product，不以小數近似替代。",
          "en": "Recompute inner parent off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "inner parent off-diagonal product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "inner parent determinant := sub(95, 4)",
      "after": "inner parent determinant = 91",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner parent determinant，不以小數近似替代。",
          "en": "Recompute inner parent determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "95",
          "4"
        ],
        "output": "91",
        "label": "inner parent determinant"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "same real R2 inner-product candidate matrix=[[19, 2], [2, 5]]",
      "after": "symmetric=是;leading=19;determinant=91",
      "relation": "implies",
      "rule_id": "innernorm_certified_same_SPD_parent",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核同一R²對稱正定父矩陣，不以無關的Euclidean或座標norm替代；對稱、首項>0、det>0才可沿用內積。",
        "en": "Qualify the same symmetric positive-definite real R² parent matrix,not an unrelated Euclidean or coordinate norm.Symmetry,positive leading entry and positive determinant are required."
      },
      "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": "inner parent completed-square slope := div(2, 19)",
      "after": "inner parent completed-square slope = 2/19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner parent completed-square slope，不以小數近似替代。",
          "en": "Recompute inner parent completed-square slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "19"
        ],
        "output": "2/19",
        "label": "inner parent completed-square slope"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "inner parent completed-square remainder := div(91, 19)",
      "after": "inner parent completed-square remainder = 91/19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner parent completed-square remainder，不以小數近似替代。",
          "en": "Recompute inner parent completed-square remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "91",
          "19"
        ],
        "output": "91/19",
        "label": "inner parent completed-square remainder"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "positive leading=19;slope=2/19;positive remainder=91/19",
      "after": "q(s,t)=(19)*(s+(2/19)*t)^2+(91/19)*t^2;positive iff(s,t)!=0",
      "relation": "implies",
      "rule_id": "innernorm_parent_global_positive_square_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "配方兩個正係數保證每個非零實向量的自內積正，零向量為0；不是只驗幾組正數值。",
        "en": "The two positive completed-square coefficients give positive self-inner-product for every nonzero real vector and0 at zero,not merely sampled positive values."
      },
      "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": "same certified real inner product;arbitrary real x,y",
      "after": "if y=0 then inner(x,y)=0;otherwise yy>0 and q(x-a*y)=xx-2*a*xy+a^2*yy>=0",
      "relation": "implies",
      "rule_id": "innernorm_Cauchy_Schwarz_all_vectors_quadratic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對任意實a，自內積非負。y非零時yy>0；取a=xy/yy得到xx−xy²/yy≥0，乘正yy得xy²≤xx*yy。y=0分開處理，不作0/0。",
        "en": "For every real a,self-inner-product is nonnegative.When y≠0,yy>0;choose a=xy/yy to obtain xx−xy²/yy≥0,multiply by positive yy to get xy²≤xx*yy.Handle y=0 separately without0/0."
      },
      "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": "norm(x)=nonnegative sqrt(inner(x,x));inner product positive definite and bilinear",
      "after": "norm(x)>=0;norm(x)=0 iff x=0;norm(a*x)=abs(a)*norm(x)",
      "relation": "implies",
      "rule_id": "innernorm_nonnegative_root_and_absolute_homogeneity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取非負根才是norm；由q(ax)=a²q(x)取根得|a|而非a，包含負純量，零向量與齊次性全稱成立。",
        "en": "Only the nonnegative root is the norm.From q(ax)=a²q(x),taking roots gives|a|,not a,including negative scalars.The zero and homogeneity claims are universal."
      },
      "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": "Cauchy-Schwarz holds for arbitrary x,y in this same inner product",
      "after": "norm(x+y)^2=xx+2*xy+yy<=(norm(x)+norm(y))^2;nonnegative roots give triangle inequality",
      "relation": "implies",
      "rule_id": "innernorm_triangle_uses_universal_Cauchy_Schwarz",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以|xy|≤norm(x)norm(y)界交叉項，再因兩側非負取根，才得到所有向量的三角責任；一次精確代值不是此全稱證書的來源，也不推出完備性。",
        "en": "Bound the cross term by|xy|≤norm(x)norm(y),then take nonnegative roots for the universal triangle inequality.One exact instance is not the theorem's source,and no completeness follows."
      },
      "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": "x self-inner-product:same matrix=[[19, 2], [2, 5]];first vector=[0, 0];second vector=[0, 0]",
      "after": "inner product=first^T*(matrix*second)",
      "relation": "implies",
      "rule_id": "innernorm_original_slot_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一父內積先算矩陣乘第二槽，再與第一槽內積；推導出的有理向量仍屬R²，不套原整數輸入界截斷。",
        "en": "Use the same parent inner product:matrix times second slot,then dot with first.Derived rational vectors still belong to R² and are not clipped to raw integer 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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "x self-inner-product matrix term0,0 := mul(19, 0)",
      "after": "x self-inner-product matrix term0,0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product matrix term0,0，不以小數近似替代。",
          "en": "Recompute x self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product matrix term0,0"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "x self-inner-product matrix term0,1 := mul(2, 0)",
      "after": "x self-inner-product matrix term0,1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product matrix term0,1，不以小數近似替代。",
          "en": "Recompute x self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product matrix term0,1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "x self-inner-product image coordinate0 := sum(0, 0)",
      "after": "x self-inner-product image coordinate0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product image coordinate0，不以小數近似替代。",
          "en": "Recompute x self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product image coordinate0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "x self-inner-product matrix term1,0 := mul(2, 0)",
      "after": "x self-inner-product matrix term1,0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product matrix term1,0，不以小數近似替代。",
          "en": "Recompute x self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product matrix term1,0"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "x self-inner-product matrix term1,1 := mul(5, 0)",
      "after": "x self-inner-product matrix term1,1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product matrix term1,1，不以小數近似替代。",
          "en": "Recompute x self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product matrix term1,1"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "x self-inner-product image coordinate1 := sum(0, 0)",
      "after": "x self-inner-product image coordinate1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product image coordinate1，不以小數近似替代。",
          "en": "Recompute x self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product image coordinate1"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "x self-inner-product first-slot term0 := mul(0, 0)",
      "after": "x self-inner-product first-slot term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product first-slot term0，不以小數近似替代。",
          "en": "Recompute x self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product first-slot term0"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "x self-inner-product first-slot term1 := mul(0, 0)",
      "after": "x self-inner-product first-slot term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product first-slot term1，不以小數近似替代。",
          "en": "Recompute x self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product first-slot term1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "x self-inner-product inner sum := sum(0, 0)",
      "after": "x self-inner-product inner sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x self-inner-product inner sum，不以小數近似替代。",
          "en": "Recompute x self-inner-product inner sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "x self-inner-product inner sum"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "y self-inner-product:same matrix=[[19, 2], [2, 5]];first vector=[1, -5];second vector=[1, -5]",
      "after": "inner product=first^T*(matrix*second)",
      "relation": "implies",
      "rule_id": "innernorm_original_slot_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一父內積先算矩陣乘第二槽，再與第一槽內積；推導出的有理向量仍屬R²，不套原整數輸入界截斷。",
        "en": "Use the same parent inner product:matrix times second slot,then dot with first.Derived rational vectors still belong to R² and are not clipped to raw integer 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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "y self-inner-product matrix term0,0 := mul(19, 1)",
      "after": "y self-inner-product matrix term0,0 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product matrix term0,0，不以小數近似替代。",
          "en": "Recompute y self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "1"
        ],
        "output": "19",
        "label": "y self-inner-product matrix term0,0"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "y self-inner-product matrix term0,1 := mul(2, -5)",
      "after": "y self-inner-product matrix term0,1 = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product matrix term0,1，不以小數近似替代。",
          "en": "Recompute y self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-5"
        ],
        "output": "-10",
        "label": "y self-inner-product matrix term0,1"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "y self-inner-product image coordinate0 := sum(19, -10)",
      "after": "y self-inner-product image coordinate0 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product image coordinate0，不以小數近似替代。",
          "en": "Recompute y self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "19",
          "-10"
        ],
        "output": "9",
        "label": "y self-inner-product image coordinate0"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "y self-inner-product matrix term1,0 := mul(2, 1)",
      "after": "y self-inner-product matrix term1,0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product matrix term1,0，不以小數近似替代。",
          "en": "Recompute y self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "y self-inner-product matrix term1,0"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "y self-inner-product matrix term1,1 := mul(5, -5)",
      "after": "y self-inner-product matrix term1,1 = -25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product matrix term1,1，不以小數近似替代。",
          "en": "Recompute y self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "-5"
        ],
        "output": "-25",
        "label": "y self-inner-product matrix term1,1"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "y self-inner-product image coordinate1 := sum(2, -25)",
      "after": "y self-inner-product image coordinate1 = -23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product image coordinate1，不以小數近似替代。",
          "en": "Recompute y self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "-25"
        ],
        "output": "-23",
        "label": "y self-inner-product image coordinate1"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "y self-inner-product first-slot term0 := mul(1, 9)",
      "after": "y self-inner-product first-slot term0 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product first-slot term0，不以小數近似替代。",
          "en": "Recompute y self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "9"
        ],
        "output": "9",
        "label": "y self-inner-product first-slot term0"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "y self-inner-product first-slot term1 := mul(-5, -23)",
      "after": "y self-inner-product first-slot term1 = 115",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product first-slot term1，不以小數近似替代。",
          "en": "Recompute y self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "-23"
        ],
        "output": "115",
        "label": "y self-inner-product first-slot term1"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "y self-inner-product inner sum := sum(9, 115)",
      "after": "y self-inner-product inner sum = 124",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y self-inner-product inner sum，不以小數近似替代。",
          "en": "Recompute y self-inner-product inner sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "115"
        ],
        "output": "124",
        "label": "y self-inner-product inner sum"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "parallelogram sum coordinate0 := sum(0, 1)",
      "after": "parallelogram sum coordinate0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 parallelogram sum coordinate0，不以小數近似替代。",
          "en": "Recompute parallelogram sum coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "parallelogram sum coordinate0"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "parallelogram sum coordinate1 := sum(0, -5)",
      "after": "parallelogram sum coordinate1 = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 parallelogram sum coordinate1，不以小數近似替代。",
          "en": "Recompute parallelogram sum coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-5"
        ],
        "output": "-5",
        "label": "parallelogram sum coordinate1"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "parallelogram difference coordinate0 := sub(0, 1)",
      "after": "parallelogram difference coordinate0 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 parallelogram difference coordinate0，不以小數近似替代。",
          "en": "Recompute parallelogram difference coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "1"
        ],
        "output": "-1",
        "label": "parallelogram difference coordinate0"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "parallelogram difference coordinate1 := sub(0, -5)",
      "after": "parallelogram difference coordinate1 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 parallelogram difference coordinate1，不以小數近似替代。",
          "en": "Recompute parallelogram difference coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-5"
        ],
        "output": "5",
        "label": "parallelogram difference coordinate1"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "sum self-inner-product:same matrix=[[19, 2], [2, 5]];first vector=[1, -5];second vector=[1, -5]",
      "after": "inner product=first^T*(matrix*second)",
      "relation": "implies",
      "rule_id": "innernorm_original_slot_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一父內積先算矩陣乘第二槽，再與第一槽內積；推導出的有理向量仍屬R²，不套原整數輸入界截斷。",
        "en": "Use the same parent inner product:matrix times second slot,then dot with first.Derived rational vectors still belong to R² and are not clipped to raw integer 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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "sum self-inner-product matrix term0,0 := mul(19, 1)",
      "after": "sum self-inner-product matrix term0,0 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product matrix term0,0，不以小數近似替代。",
          "en": "Recompute sum self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "1"
        ],
        "output": "19",
        "label": "sum self-inner-product matrix term0,0"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "sum self-inner-product matrix term0,1 := mul(2, -5)",
      "after": "sum self-inner-product matrix term0,1 = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product matrix term0,1，不以小數近似替代。",
          "en": "Recompute sum self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-5"
        ],
        "output": "-10",
        "label": "sum self-inner-product matrix term0,1"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "sum self-inner-product image coordinate0 := sum(19, -10)",
      "after": "sum self-inner-product image coordinate0 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product image coordinate0，不以小數近似替代。",
          "en": "Recompute sum self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "19",
          "-10"
        ],
        "output": "9",
        "label": "sum self-inner-product image coordinate0"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "sum self-inner-product matrix term1,0 := mul(2, 1)",
      "after": "sum self-inner-product matrix term1,0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product matrix term1,0，不以小數近似替代。",
          "en": "Recompute sum self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "sum self-inner-product matrix term1,0"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "sum self-inner-product matrix term1,1 := mul(5, -5)",
      "after": "sum self-inner-product matrix term1,1 = -25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product matrix term1,1，不以小數近似替代。",
          "en": "Recompute sum self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "-5"
        ],
        "output": "-25",
        "label": "sum self-inner-product matrix term1,1"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "sum self-inner-product image coordinate1 := sum(2, -25)",
      "after": "sum self-inner-product image coordinate1 = -23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product image coordinate1，不以小數近似替代。",
          "en": "Recompute sum self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "-25"
        ],
        "output": "-23",
        "label": "sum self-inner-product image coordinate1"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "sum self-inner-product first-slot term0 := mul(1, 9)",
      "after": "sum self-inner-product first-slot term0 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product first-slot term0，不以小數近似替代。",
          "en": "Recompute sum self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "9"
        ],
        "output": "9",
        "label": "sum self-inner-product first-slot term0"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "sum self-inner-product first-slot term1 := mul(-5, -23)",
      "after": "sum self-inner-product first-slot term1 = 115",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product first-slot term1，不以小數近似替代。",
          "en": "Recompute sum self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "-23"
        ],
        "output": "115",
        "label": "sum self-inner-product first-slot term1"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "sum self-inner-product inner sum := sum(9, 115)",
      "after": "sum self-inner-product inner sum = 124",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum self-inner-product inner sum，不以小數近似替代。",
          "en": "Recompute sum self-inner-product inner sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "115"
        ],
        "output": "124",
        "label": "sum self-inner-product inner sum"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "difference self-inner-product:same matrix=[[19, 2], [2, 5]];first vector=[-1, 5];second vector=[-1, 5]",
      "after": "inner product=first^T*(matrix*second)",
      "relation": "implies",
      "rule_id": "innernorm_original_slot_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一父內積先算矩陣乘第二槽，再與第一槽內積；推導出的有理向量仍屬R²，不套原整數輸入界截斷。",
        "en": "Use the same parent inner product:matrix times second slot,then dot with first.Derived rational vectors still belong to R² and are not clipped to raw integer 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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "difference self-inner-product matrix term0,0 := mul(19, -1)",
      "after": "difference self-inner-product matrix term0,0 = -19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product matrix term0,0，不以小數近似替代。",
          "en": "Recompute difference self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "-1"
        ],
        "output": "-19",
        "label": "difference self-inner-product matrix term0,0"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "difference self-inner-product matrix term0,1 := mul(2, 5)",
      "after": "difference self-inner-product matrix term0,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product matrix term0,1，不以小數近似替代。",
          "en": "Recompute difference self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "5"
        ],
        "output": "10",
        "label": "difference self-inner-product matrix term0,1"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "difference self-inner-product image coordinate0 := sum(-19, 10)",
      "after": "difference self-inner-product image coordinate0 = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product image coordinate0，不以小數近似替代。",
          "en": "Recompute difference self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-19",
          "10"
        ],
        "output": "-9",
        "label": "difference self-inner-product image coordinate0"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "difference self-inner-product matrix term1,0 := mul(2, -1)",
      "after": "difference self-inner-product matrix term1,0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product matrix term1,0，不以小數近似替代。",
          "en": "Recompute difference self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "-2",
        "label": "difference self-inner-product matrix term1,0"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "difference self-inner-product matrix term1,1 := mul(5, 5)",
      "after": "difference self-inner-product matrix term1,1 = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product matrix term1,1，不以小數近似替代。",
          "en": "Recompute difference self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "5"
        ],
        "output": "25",
        "label": "difference self-inner-product matrix term1,1"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "difference self-inner-product image coordinate1 := sum(-2, 25)",
      "after": "difference self-inner-product image coordinate1 = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product image coordinate1，不以小數近似替代。",
          "en": "Recompute difference self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "25"
        ],
        "output": "23",
        "label": "difference self-inner-product image coordinate1"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "difference self-inner-product first-slot term0 := mul(-1, -9)",
      "after": "difference self-inner-product first-slot term0 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product first-slot term0，不以小數近似替代。",
          "en": "Recompute difference self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-9"
        ],
        "output": "9",
        "label": "difference self-inner-product first-slot term0"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "difference self-inner-product first-slot term1 := mul(5, 23)",
      "after": "difference self-inner-product first-slot term1 = 115",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product first-slot term1，不以小數近似替代。",
          "en": "Recompute difference self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "23"
        ],
        "output": "115",
        "label": "difference self-inner-product first-slot term1"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "difference self-inner-product inner sum := sum(9, 115)",
      "after": "difference self-inner-product inner sum = 124",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference self-inner-product inner sum，不以小數近似替代。",
          "en": "Recompute difference self-inner-product inner sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "115"
        ],
        "output": "124",
        "label": "difference self-inner-product inner sum"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "parallelogram left sum := sum(124, 124)",
      "after": "parallelogram left sum = 248",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 parallelogram left sum，不以小數近似替代。",
          "en": "Recompute parallelogram left sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "124",
          "124"
        ],
        "output": "248",
        "label": "parallelogram left sum"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "twice x self-inner-product := mul(2, 0)",
      "after": "twice x self-inner-product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 twice x self-inner-product，不以小數近似替代。",
          "en": "Recompute twice x self-inner-product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "twice x self-inner-product"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "twice y self-inner-product := mul(2, 124)",
      "after": "twice y self-inner-product = 248",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 twice y self-inner-product，不以小數近似替代。",
          "en": "Recompute twice y self-inner-product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "124"
        ],
        "output": "248",
        "label": "twice y self-inner-product"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "parallelogram right sum := sum(0, 248)",
      "after": "parallelogram right sum = 248",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 parallelogram right sum，不以小數近似替代。",
          "en": "Recompute parallelogram right sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "248"
        ],
        "output": "248",
        "label": "parallelogram right sum"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "same inner product;left=248;right=248",
      "after": "bilinear expansions have opposite cross terms which cancel;identity=是",
      "relation": "implies",
      "rule_id": "parallelogram_same_form_identity_not_Hilbert_claim",
      "conditions": [],
      "explanation": {
        "zh-Hant": "展開q(x+y)與q(x−y)，交叉項±2xy相消，剩2q(x)+2q(y)；不是從這次相等反向建構內積，也不是Hilbert完備證書。",
        "en": "Expand q(x+y),q(x−y);their opposite±2xy terms cancel to2q(x)+2q(y).This instance neither constructs an inner product by converse nor certifies Hilbert completeness."
      },
      "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": [
      [
        0,
        0
      ],
      [
        1,
        -5
      ],
      248,
      248,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 0], [1, -5], 248, 248, 是)",
    "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-20261029-r03-inner-product-induced-norm-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations/questions.jsonl",
      "batch_sha256": "d36475f58ad7565313bbc6c778af6c9218c8403f668536cfa36f61eb2d99f0b2",
      "original_line": 93,
      "original_line_sha256": "c2c4b791f1cc40e7ae5e86065022121d318f3c2d60937e64d521d44d740c6830",
      "question_compact_sha256": "2c13fc1e646bf7f13de812d5bfbee5d1528124c92f884854cce8be65ea7a9d84",
      "english_row_compact_sha256": "961ba066159d67c67649eec4c2a0ea6449fddc8fd8d958610c3bdfbc649249cc",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261029-r03-inner-product-induced-norm-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-inner-product-induced-norm": {
        "sha256": "9b9498dd5a3aa838a1854d97c526a01a657f0f88bab64b00a78ef20fb1f80c97",
        "english_sha256": "ca3df586cdb1653a3796cb144948a8b5f7eb9a7d8b9950440e12d784e2757548",
        "original_path": "public/data/mko/inner-product-induced-norm.json",
        "english_path": "public/data/i18n/editions/mko-inner-product-induced-norm.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "466f72389eb454e24322e59df2871ec206c351545a4a7b072d39fb5f6afc2697",
    "original_candidate_sha256": "10e5da66471287ee736f30b5540758f07a2bfb4d1a3b5671f67f9e936aba9521",
    "original_candidate_line_sha256": "10e5da66471287ee736f30b5540758f07a2bfb4d1a3b5671f67f9e936aba9521"
  },
  "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-p29",
    "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": "bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae",
      "question_first_witness_sha256": "61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 59,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4310a1f5957e86775712ae8188d9c35a430a006470a3029fbe74ea26be22764c",
      "arithmetic_steps": 49
    },
    "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."
  }
}