{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r06-partial-differential-equation-0057",
  "pilot_number": "R0022-11657",
  "number": 11657,
  "family_id": "qf-mod24-pde-principal-classification",
  "title": {
    "zh-Hant": "第 11657 題：線性PDE切片",
    "en": "Question 11657: Classify Only after Fixing b*u_xy rather than 2*b*u_xy"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r06-partial-differential-equation-0057",
    "batch_id": "batch-20261013-advance-20261024-r06-partial-differential-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-partial-differential-equation"
    ],
    "title_zh": "線性PDE切片",
    "stem_zh": "明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,-6,3]}。請依 [principal,convention,discriminant,classification] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)",
      "explanation_zh": "依賴結構核對=([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "線性PDE切片"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-pde-principal-classification",
      "seed": 202610246057,
      "parameters": {
        "principal": [
          1,
          -6,
          3
        ],
        "domain": "Polynomial_linear_PDE_operator_slice",
        "operation": "pde-principal-classification",
        "semantic_template": "qf-mod24-pde-principal-classification/{\"principal\":[1,-6,3]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261024-r06-partial-differential-equation-0057",
    "batch_id": "batch-20261013-advance-20261024-r06-partial-differential-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "47f967d58c2832ac1f6a687820bd97260cb81af441d8ca04fb6eca21a859741d",
      "family_id": "qf-mod24-pde-principal-classification",
      "seed": 202610246057,
      "semantic_template": "qf-mod24-pde-principal-classification/{\"principal\":[1,-6,3]}"
    },
    "fields": {
      "title": "Classify Only after Fixing b*u_xy rather than 2*b*u_xy",
      "stem": "Use the nonzero second-order principal part [a,b,c]=[1, -6, 3] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.",
      "answer": {
        "kind": "expression",
        "value": "([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)",
        "explanation": "Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)."
      },
      "skills": [
        "Classify Only after Fixing b*u_xy rather than 2*b*u_xy"
      ],
      "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": "明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,-6,3]}。請依 [principal,convention,discriminant,classification] 欄序回答。",
        "en": "Use the nonzero second-order principal part [a,b,c]=[1, -6, 3] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "principal mixed coefficient squared := mul(-6, -6)",
      "after": "principal mixed coefficient squared = 36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 principal mixed coefficient squared，不以小數近似替代。",
          "en": "Recompute principal mixed coefficient squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-6",
          "-6"
        ],
        "output": "36",
        "label": "principal mixed coefficient squared"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "principal a times c := mul(1, 3)",
      "after": "principal a times c = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 principal a times c，不以小數近似替代。",
          "en": "Recompute principal a times c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "principal a times c"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "principal four times ac := mul(4, 3)",
      "after": "principal four times ac = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 principal four times ac，不以小數近似替代。",
          "en": "Recompute principal four times ac with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "3"
        ],
        "output": "12",
        "label": "principal four times ac"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "principal discriminant := sub(36, 12)",
      "after": "principal discriminant = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 principal discriminant，不以小數近似替代。",
          "en": "Recompute principal discriminant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "36",
          "12"
        ],
        "output": "24",
        "label": "principal discriminant"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "nonzero principal=[1, -6, 3];operator=a*u_xx+b*u_xy+c*u_yy",
      "after": "principal symbol=a*xi^2+b*xi*eta+c*eta^2;discriminant=24;classification=hyperbolic",
      "relation": "implies",
      "rule_id": "principal_symbol_mixed_convention_classification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "主符號的對稱矩陣交叉係數是b/2，因此其行列式為ac−b²/4。D>0為不定、D<0為定、D=0為退化，按宣告慣例分類；全零主部排除，分類本身不保證適定性。",
        "en": "The symmetric principal-symbol matrix has off-diagonal b/2,hence determinant ac−b²/4.D>0 gives an indefinite symbol,D<0 a definite one,and D=0 a degenerate one,under the declared convention.The all-zero principal part is excluded;classification alone gives no well-posedness."
      },
      "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": [
      [
        1,
        -6,
        3
      ],
      "a*u_xx+b*u_xy+c*u_yy",
      24,
      "hyperbolic"
    ]
  },
  "conclusion": {
    "canonical_value": "([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)",
    "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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        "english_path": "public/data/i18n/editions/mko-partial-differential-equation.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
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    "original_candidate_sha256": "a1e517eb64c751940a73b05688ece156eae62ab066222b5709b5515c8f01523a",
    "original_candidate_line_sha256": "a1e517eb64c751940a73b05688ece156eae62ab066222b5709b5515c8f01523a"
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      "formalization": "not_formalized",
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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."
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  "revision": {
    "id": "review-20261026-006-p23",
    "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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    "bilingual_semantics": true,
    "steps_reviewed": 5,
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    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}