{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261027-r04-real-bilinear-form-0091",
  "pilot_number": "R0024-12791",
  "number": 12791,
  "family_id": "qf-mod27-bilinear-symmetry-audit",
  "title": {
    "zh-Hant": "第 12791 題：實雙線性形式",
    "en": "Question 12791: Symmetry Is Additional to Bilinearity"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261027-r04-real-bilinear-form-0091",
    "batch_id": "batch-20261014-advance-20261027-r04-real-bilinear-form-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-real-bilinear-form"
    ],
    "title_zh": "實雙線性形式",
    "stem_zh": "雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[3,-2],[4,2]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[3, -2], [4, 2]], [0, 1, -2, 4], 否, symmetry_is_extra_not_bilinearity)",
      "explanation_zh": "Wave4根節點核對=([[3, -2], [4, 2]], [0, 1, -2, 4], 否, symmetry_is_extra_not_bilinearity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "實雙線性形式"
      ]
    },
    "variant": {
      "family_id": "qf-mod27-bilinear-symmetry-audit",
      "seed": 202610274091,
      "parameters": {
        "matrix": [
          [
            3,
            -2
          ],
          [
            4,
            2
          ]
        ],
        "domain": "Exact_matrix_real_bilinear_form",
        "operation": "bilinear-symmetry-audit",
        "semantic_template": "qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[3,-2],[4,2]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261027-r04-real-bilinear-form-0091",
    "batch_id": "batch-20261014-advance-20261027-r04-real-bilinear-form-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "45d5a2949a98ec51cef0f275661171a85d45e6e986912ae56343e3061f5f4ddc",
      "family_id": "qf-mod27-bilinear-symmetry-audit",
      "seed": 202610274091,
      "semantic_template": "qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[3,-2],[4,2]]}"
    },
    "fields": {
      "title": "Symmetry Is Additional to Bilinearity",
      "stem": "Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).",
      "answer": {
        "kind": "expression",
        "value": "([[3, -2], [4, 2]], [0, 1, -2, 4], No, symmetry_is_extra_not_bilinearity)",
        "explanation": "Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, -2], [4, 2]], [0, 1, -2, 4], No, symmetry_is_extra_not_bilinearity)."
      },
      "skills": [
        "Symmetry Is Additional to Bilinearity"
      ],
      "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": "雙線性必須分別核第一槽與第二槽。操作=bilinear-symmetry-audit；資料={\"matrix\":[[3,-2],[4,2]]}。請依 [matrix,first_mismatch,symmetric,boundary] 欄序回答。",
        "en": "Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "fixed real matrix=[[3, -2], [4, 2]];B(x,y)=sum_i sum_j x_i*A_ij*y_j",
      "after": "distributivity proves real linearity in each slot separately for all vectors and scalars",
      "relation": "implies",
      "rule_id": "matrix_formula_bilinearity_universal_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限雙重和中實數分配律可把任意第一槽或第二槽線性組合展開，得到各自的全稱線性證書；整數實例只是此固定公式的求值，不是黑盒抽樣證明。",
        "en": "Distributivity expands any real linear combination in either slot of the finite double sum,giving separate universal linearity certificates.Integer instances evaluate this fixed formula,not prove a black box by sampling."
      },
      "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": "matrix coordinates=[0, 1];A_ij=-2;A_ji=4",
      "after": "opposite entries equal=否",
      "relation": "implies",
      "rule_id": "bilinear_each_opposite_matrix_entry",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用標準基向量e_i、e_j，B(e_i,e_j)=Aij，因此不相等立即給對稱性反例；完整相等才有A=A轉置的全稱理由。",
        "en": "Standard basis vectors give B(e_i,e_j)=A_ij,so a mismatch immediately refutes symmetry.All opposite entries must agree for the universal A=A^T certificate."
      },
      "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": "first mismatch in row order=[0, 1, -2, 4]",
      "after": "symmetric=否;symmetry_is_extra_not_bilinearity",
      "relation": "implies",
      "rule_id": "bilinear_symmetry_additional_not_inner_product",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若A=A轉置，雙重和交換索引給B(x,y)=B(y,x)對所有實向量成立。對稱仍不代表正定、非退化或內積，例如零矩陣。",
        "en": "If A=A^T,exchanging indices in the double sum gives symmetry for all real vectors.Symmetry still does not imply positive definiteness,nondegeneracy or an inner product;the zero matrix is an example."
      },
      "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": [
      [
        [
          3,
          -2
        ],
        [
          4,
          2
        ]
      ],
      [
        0,
        1,
        -2,
        4
      ],
      false,
      "symmetry_is_extra_not_bilinearity"
    ]
  },
  "conclusion": {
    "canonical_value": "([[3, -2], [4, 2]], [0, 1, -2, 4], 否, symmetry_is_extra_not_bilinearity)",
    "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."
  },
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    "original_candidate_line_sha256": "331ba46d7dd4d0fec752240b6fb7aa834973b9b0661dc3ceb0ad3cd85d0a0a92"
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      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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    "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": {
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    "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.",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}