{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261028-r04-real-inner-product-space-0090",
  "pilot_number": "R0026-13865",
  "number": 13865,
  "family_id": "qf-mod28-inner-qualification",
  "title": {
    "zh-Hant": "第 13865 題：實內積空間",
    "en": "Question 13865: Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261028-r04-real-inner-product-space-0090",
    "batch_id": "batch-20261014-advance-20261028-r04-real-inner-product-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-real-inner-product-space"
    ],
    "title_zh": "實內積空間",
    "stem_zh": "內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[15,0],[0,0]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[15, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)",
      "explanation_zh": "Wave4第二層核對=([[15, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；R2正定使用Sylvester全稱證書；半正定或有限向量樣本不足且不推出完備。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "實內積空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod28-inner-qualification",
      "seed": 202610284090,
      "parameters": {
        "matrix": [
          [
            15,
            0
          ],
          [
            0,
            0
          ]
        ],
        "domain": "Exact_R2_matrix_inner_product",
        "operation": "inner-qualification",
        "semantic_template": "qf-mod28-inner-qualification/{\"matrix\":[[15,0],[0,0]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261028-r04-real-inner-product-space-0090",
    "batch_id": "batch-20261014-advance-20261028-r04-real-inner-product-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ec0dcca46d5f8c306bddf855182d111d974a2d7f51e67228a47d4c6e1780a1e3",
      "family_id": "qf-mod28-inner-qualification",
      "seed": 202610284090,
      "semantic_template": "qf-mod28-inner-qualification/{\"matrix\":[[15,0],[0,0]]}"
    },
    "fields": {
      "title": "Conjoin Bilinearity, Symmetry and Strict Positive Definiteness",
      "stem": "Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.",
      "answer": {
        "kind": "expression",
        "value": "([[15, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)",
        "explanation": "The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)."
      },
      "skills": [
        "Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"
      ],
      "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": "內積須同時雙線性對稱且正定。操作=inner-qualification；資料={\"matrix\":[[15,0],[0,0]]}。請依 [matrix,bilinear,symmetric,positive_definite,inner_product,classification] 欄序回答。",
        "en": "Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "candidate real R2 form;matrix=[[15, 0], [0, 0]];B(x,y)=x^T*A*y",
      "after": "matrix distributivity supplies bilinearity;other inner-product obligations remain separate",
      "relation": "implies",
      "rule_id": "inner_candidate_bilinear_not_presumed_inner_product",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩陣公式本身雙線性，但不先假設對稱或正定；本協定只處理實數R²，不加複共軛或一般維度推論。",
        "en": "The matrix formula is bilinear without presuming symmetry or positive definiteness.This protocol is real R²,not a complex-conjugate or arbitrary-dimensional claim."
      },
      "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": "opposite off-diagonal entries=[0, 0]",
      "after": "symmetric=是",
      "relation": "implies",
      "rule_id": "inner_symmetry_additional_to_bilinearity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以標準基向量可讀出兩側矩陣係數，因此對稱當且僅當兩個非對角係數一致；不能由一次B(x,y)=B(y,x)猜全稱對稱。",
        "en": "Standard basis vectors read off matrix entries,so symmetry is equivalent to equality of the two off-diagonal entries.One pairwise equality cannot establish universal symmetry."
      },
      "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": "inner determinant diagonal product := mul(15, 0)",
      "after": "inner determinant diagonal product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner determinant diagonal product，不以小數近似替代。",
          "en": "Recompute inner determinant diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "0"
        ],
        "output": "0",
        "label": "inner determinant diagonal product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "inner determinant off-diagonal product := mul(0, 0)",
      "after": "inner determinant off-diagonal product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner determinant off-diagonal product，不以小數近似替代。",
          "en": "Recompute inner determinant off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "inner determinant off-diagonal product"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "inner signed determinant := sub(0, 0)",
      "after": "inner signed determinant = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner signed determinant，不以小數近似替代。",
          "en": "Recompute inner signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "inner signed determinant"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "inner completed-square slope := div(0, 15)",
      "after": "inner completed-square slope = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner completed-square slope，不以小數近似替代。",
          "en": "Recompute inner completed-square slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "15"
        ],
        "output": "0",
        "label": "inner completed-square slope"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "inner completed-square remainder := div(0, 15)",
      "after": "inner completed-square remainder = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inner completed-square remainder，不以小數近似替代。",
          "en": "Recompute inner completed-square remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "15"
        ],
        "output": "0",
        "label": "inner completed-square remainder"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "symmetric matrix=[[15, 0], [0, 0]];positive leading=15;b/a=0;det/a=0",
      "after": "q(s,t)=(15)*(s+(0)*t)^2+(0)*t^2",
      "relation": "implies",
      "rule_id": "R2_Sylvester_completed_square_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在a>0才配方。若det>0，t非零時第二項正，t=0且s非零時第一項正；反向嚴格正定先由e1得a>0，再令s=−(b/a)t、t=1得det/a>0。",
        "en": "Complete the square only after a>0.If det>0,the second term is positive for t≠0;when t=0 and s≠0,the first is positive.Conversely strict positivity first gives a>0 at e1,then det/a>0 at s=−b/a,t=1."
      },
      "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": "nonpositive determinant witness matrix second slot:matrix=[[15, 0], [0, 0]];input=[0, 15]",
      "after": "input dimension=2;output dimension=2;rows dot input",
      "relation": "implies",
      "rule_id": "matrix_row_column_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留長方形矩陣的欄輸入、列輸出角色，計算值可超過原輸入座標界，不重新截斷。",
        "en": "Retain column-input/row-output roles for rectangular matrices.Computed values can exceed raw input bounds and are not clipped."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "nonpositive determinant witness matrix second slot row0 column0 product := mul(15, 0)",
      "after": "nonpositive determinant witness matrix second slot row0 column0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness matrix second slot row0 column0 product，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness matrix second slot row0 column0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "0"
        ],
        "output": "0",
        "label": "nonpositive determinant witness matrix second slot row0 column0 product"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "nonpositive determinant witness matrix second slot row0 column1 product := mul(0, 15)",
      "after": "nonpositive determinant witness matrix second slot row0 column1 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness matrix second slot row0 column1 product，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness matrix second slot row0 column1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "15"
        ],
        "output": "0",
        "label": "nonpositive determinant witness matrix second slot row0 column1 product"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "nonpositive determinant witness matrix second slot row0 sum := sum(0, 0)",
      "after": "nonpositive determinant witness matrix second slot row0 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness matrix second slot row0 sum，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness matrix second slot row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "nonpositive determinant witness matrix second slot row0 sum"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "nonpositive determinant witness matrix second slot row1 column0 product := mul(0, 0)",
      "after": "nonpositive determinant witness matrix second slot row1 column0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness matrix second slot row1 column0 product，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness matrix second slot row1 column0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "nonpositive determinant witness matrix second slot row1 column0 product"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "nonpositive determinant witness matrix second slot row1 column1 product := mul(0, 15)",
      "after": "nonpositive determinant witness matrix second slot row1 column1 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness matrix second slot row1 column1 product，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness matrix second slot row1 column1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "15"
        ],
        "output": "0",
        "label": "nonpositive determinant witness matrix second slot row1 column1 product"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "nonpositive determinant witness matrix second slot row1 sum := sum(0, 0)",
      "after": "nonpositive determinant witness matrix second slot row1 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness matrix second slot row1 sum，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness matrix second slot row1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "nonpositive determinant witness matrix second slot row1 sum"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "nonpositive determinant witness first coordinate0 product := mul(0, 0)",
      "after": "nonpositive determinant witness first coordinate0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness first coordinate0 product，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness first coordinate0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "nonpositive determinant witness first coordinate0 product"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "nonpositive determinant witness first coordinate1 product := mul(15, 0)",
      "after": "nonpositive determinant witness first coordinate1 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness first coordinate1 product，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness first coordinate1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "0"
        ],
        "output": "0",
        "label": "nonpositive determinant witness first coordinate1 product"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "nonpositive determinant witness bilinear sum := sum(0, 0)",
      "after": "nonpositive determinant witness bilinear sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色與已核中間量精確計算；保留座標、係數、正負號及必要條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonpositive determinant witness bilinear sum，不以小數近似替代。",
          "en": "Recompute nonpositive determinant witness bilinear sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "nonpositive determinant witness bilinear sum"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "nonzero witness=[0, 15];q(witness)=0=leading*determinant",
      "after": "strict positivity fails;zero determinant is not strict positive definiteness",
      "relation": "implies",
      "rule_id": "nonpositive_determinant_actual_countervector",
      "conditions": [],
      "explanation": {
        "zh-Hant": "a>0保證此向量非零；det≤0使其二次值≤0，包含半正定的零值失敗，不能只用幾個正樣本補救。",
        "en": "Positive a makes this vector nonzero;det≤0 gives a nonpositive value,including the zero-value failure of semidefiniteness.Positive samples do not repair it."
      },
      "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": "symmetric=是;leading=15;determinant=0",
      "after": "strict symmetric R2 certificate=否",
      "relation": "implies",
      "rule_id": "strict_Sylvester_conditions_conjunction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "最後旗標合取對稱、首主子式>0與行列式>0；這是此R²對稱內積協定，不額外宣稱完備。",
        "en": "Conjoin symmetry,positive leading minor and positive determinant.This is the stated symmetric R² inner-product protocol,with no additional completeness claim."
      },
      "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": [
      [
        [
          15,
          0
        ],
        [
          0,
          0
        ]
      ],
      true,
      true,
      false,
      false,
      "not_inner_product"
    ]
  },
  "conclusion": {
    "canonical_value": "([[15, 0], [0, 0]], 是, 是, 否, 否, not_inner_product)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261028-r04-real-inner-product-space-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261028-r04-real-inner-product-space-foundations/questions.jsonl",
      "batch_sha256": "25e4359f40a20ac5a02eee81479d8c960a481135e709654243fce3633ed5b98e",
      "original_line": 90,
      "original_line_sha256": "bf59b13fcb4da14db963c1634867818c9d303661b6a8b267523b346d41e8f058",
      "question_compact_sha256": "ec0dcca46d5f8c306bddf855182d111d974a2d7f51e67228a47d4c6e1780a1e3",
      "english_row_compact_sha256": "8f4d0a05d13add8e7b6afc9ef2b364148f17cb445d34ce40e3b21ec217aa303b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261028-r04-real-inner-product-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261028-r04-real-inner-product-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-real-inner-product-space": {
        "sha256": "70d1a304624117e0e814b5b9ca816a5b851798e54cc213f1f2b3460e6450f963",
        "english_sha256": "37a8aec17484c89c814b26ce16792d6a4b33d8d0ef4fffb7e9e9e05e3ec4c42f",
        "original_path": "public/data/mko/real-inner-product-space.json",
        "english_path": "public/data/i18n/editions/mko-real-inner-product-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e9c0ae2888b9f0f177728712cf7eb267a13a2a12f233a4c64ea50cb104f58bb0",
    "original_candidate_sha256": "a2dd550ccff6beea9d5621438321f621f45e8c229be11c11ea8569dab52eea71",
    "original_candidate_line_sha256": "a2dd550ccff6beea9d5621438321f621f45e8c229be11c11ea8569dab52eea71"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-007-p27",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798",
      "question_first_witness_sha256": "cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 20,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "db9e22eade3534f5730fc45e409660d2cafaa20564243a6e71815ba4c204bb8e",
      "arithmetic_steps": 14
    },
    "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."
  }
}