{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r06-partial-differential-equation-0038",
  "pilot_number": "R0022-11688",
  "number": 11688,
  "family_id": "qf-mod24-pde-second-order-point-residual",
  "title": {
    "zh-Hant": "第 11688 題：線性PDE切片",
    "en": "Question 11688: Second-Order Residual with an Explicit Mixed-Term Convention"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r06-partial-differential-equation-0038",
    "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-second-order-point-residual；資料={\"coefficients\":[0,0,2,1,-1,0],\"principal\":[1,0,1],\"source\":0,\"x\":0,\"y\":2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, 否)",
      "explanation_zh": "依賴結構核對=([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, 否)；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": 2,
      "skills": [
        "線性PDE切片"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-pde-second-order-point-residual",
      "seed": 202610246038,
      "parameters": {
        "coefficients": [
          0,
          0,
          2,
          1,
          -1,
          0
        ],
        "principal": [
          1,
          0,
          1
        ],
        "source": 0,
        "x": 0,
        "y": 2,
        "domain": "Polynomial_linear_PDE_operator_slice",
        "operation": "pde-second-order-point-residual",
        "semantic_template": "qf-mod24-pde-second-order-point-residual/{\"coefficients\":[0,0,2,1,-1,0],\"principal\":[1,0,1],\"source\":0,\"x\":0,\"y\":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、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-0038",
    "batch_id": "batch-20261013-advance-20261024-r06-partial-differential-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f4951af5a4df5dec28ddf6629b976d84ff8e877cd400005eae8049ea09dd9b59",
      "family_id": "qf-mod24-pde-second-order-point-residual",
      "seed": 202610246038,
      "semantic_template": "qf-mod24-pde-second-order-point-residual/{\"coefficients\":[0,0,2,1,-1,0],\"principal\":[1,0,1],\"source\":0,\"x\":0,\"y\":2}"
    },
    "fields": {
      "title": "Second-Order Residual with an Explicit Mixed-Term Convention",
      "stem": "Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[0, 0, 2, 1, -1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, 0, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=0, x=0, y=2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).",
      "answer": {
        "kind": "expression",
        "value": "([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, No)",
        "explanation": "Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, No)."
      },
      "skills": [
        "Second-Order Residual with an Explicit Mixed-Term Convention"
      ],
      "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-second-order-point-residual；資料={\"coefficients\":[0,0,2,1,-1,0],\"principal\":[1,0,1],\"source\":0,\"x\":0,\"y\":2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。",
        "en": "Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[0, 0, 2, 1, -1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, 0, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=0, x=0, y=2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "original coefficients=[0, 0, 2, 1, -1, 0];basis=[1,x,y,x^2,xy,y^2]",
      "after": "u(x,y)=(0)+(0)*x+(2)*y+(1)*x^2+(-1)*x*y+(0)*y^2",
      "relation": "implies",
      "rule_id": "bivariate_scalar_polynomial_basis_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "六個係數按固定單項式順序讀，不與單變量二次式的[a,b,c]降冪次序混用。",
        "en": "Read the six coefficients in their fixed monomial order,not the descending[a,b,c] order of a univariate quadratic."
      },
      "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": "u_xx constant := mul(2, 1)",
      "after": "u_xx constant = 2",
      "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 u_xx constant，不以小數近似替代。",
          "en": "Recompute u_xx constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "u_xx constant"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "u_yy constant := mul(2, 0)",
      "after": "u_yy constant = 0",
      "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 u_yy constant，不以小數近似替代。",
          "en": "Recompute u_yy constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "u_yy constant"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "u coefficients=[0, 0, 2, 1, -1, 0];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y",
      "after": "u_xx=2;u_xy=-1;u_yy=0",
      "relation": "implies",
      "rule_id": "polynomial_second_partials_explicit_monomials",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以固定另一座標的切片規則逐項微分：x的一次項再微分成常數，xy交叉項只貢獻c4；這是多項式恆等式，不是由一般混偏導交換假設。",
        "en": "Differentiate each monomial by fixed-coordinate slice rules.The linear x term differentiates to a constant,and xy contributes c4 to u_xy.These are polynomial identities,not an assumed general mixed-partial exchange theorem."
      },
      "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": "principal=[1, 0, 1];point=[0, 2]",
      "after": "operator=a*u_xx+b*u_xy+c*u_yy;not 2*b*u_xy",
      "relation": "implies",
      "rule_id": "second_order_declared_mixed_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定混合項就是b*u_xy，不擅自改成常見另一慣例2b*u_xy；即使二階值在本多項式為常數，也保留題目的點角色。",
        "en": "The mixed term is b*u_xy,not the alternative convention2b*u_xy.Retain the query point roles even though these second derivatives are constant for this polynomial."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "second order term0 := mul(1, 2)",
      "after": "second order term0 = 2",
      "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 second order term0，不以小數近似替代。",
          "en": "Recompute second order term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "second order term0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "second order term1 := mul(0, -1)",
      "after": "second order term1 = 0",
      "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 second order term1，不以小數近似替代。",
          "en": "Recompute second order term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "second order term1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "second order term2 := mul(1, 0)",
      "after": "second order term2 = 0",
      "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 second order term2，不以小數近似替代。",
          "en": "Recompute second order term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "second order term2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "second order operator sum := sum(2, 0, 0)",
      "after": "second order operator sum = 2",
      "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 second order operator sum，不以小數近似替代。",
          "en": "Recompute second order operator sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "0",
          "0"
        ],
        "output": "2",
        "label": "second order operator sum"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "second order signed residual := sub(2, 0)",
      "after": "second order signed residual = 2",
      "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 second order signed residual，不以小數近似替代。",
          "en": "Recompute second order signed residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "second order signed residual"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "query point=[0, 2];operator value=2;source=0;signed residual=2",
      "after": "satisfied at declared point=否;no general solution claim",
      "relation": "implies",
      "rule_id": "PDE_point_obligation_not_global_solution",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此欄回答指定點殘差是否為零。單點通過通常不證全域解，也不給解存在唯一、正則或適定性；全域恆等需另核完整符號殘差。",
        "en": "This flag tests the residual at the declared point.A point pass does not generally prove a global solution or existence,uniqueness,regularity or well-posedness.A global identity requires a separate full symbolic residual audit."
      },
      "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,
        2,
        1,
        -1,
        0
      ],
      [
        1,
        0,
        1
      ],
      0,
      2,
      2,
      -1,
      0,
      0,
      2,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, 否)",
    "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-20261013-advance-20261024-r06-partial-differential-equation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261024-r06-partial-differential-equation-foundations/questions.jsonl",
      "batch_sha256": "3694f95d9344a162d98b9c0854b262ae4f31f1b65ec74f3c45f754d256e09dea",
      "original_line": 38,
      "original_line_sha256": "a1d015a4486ac2eec38a8d1d55ac8e08d43afe0a735f1c1e9961969e61a51629",
      "question_compact_sha256": "f4951af5a4df5dec28ddf6629b976d84ff8e877cd400005eae8049ea09dd9b59",
      "english_row_compact_sha256": "43f40fbd0daf353b732a20570deef97c33eba7e9d2998a3fc852191e94b1ad7f",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261024-r06-partial-differential-equation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261024-r06-partial-differential-equation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-partial-differential-equation": {
        "sha256": "b78ef1b1a613523b784d17e0507361cf263e495850e2b7aa880558f1294758b8",
        "english_sha256": "62e82ace9246a9a6a884ee7e0a1a332257e56eba985caac8dbc5c8d5efa7df75",
        "original_path": "public/data/mko/partial-differential-equation.json",
        "english_path": "public/data/i18n/editions/mko-partial-differential-equation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "844102be97423689e3a5e0c418b86b56773cf737eab263267081710eff80cbea",
    "original_candidate_sha256": "935cb12ceeda6e06edd7bd96904697a87699d87d090b59e89b9591ec3c1fef89",
    "original_candidate_line_sha256": "935cb12ceeda6e06edd7bd96904697a87699d87d090b59e89b9591ec3c1fef89"
  },
  "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-20261026-006-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.",
    "review_receipts": {
      "semantic_review_sha256": "94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac",
      "question_first_witness_sha256": "0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "83e0c14a8228ee891dbc35bba889691686798b8337eba98ca45897ec3232f6d2",
      "arithmetic_steps": 7
    },
    "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."
  }
}