{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r07-eigenvalue-0072",
  "pilot_number": "R0006-03797",
  "number": 3797,
  "family_id": "qf-density-eigen-kernel-residual",
  "title": {
    "zh-Hant": "第 3797 題：指定核方程的有序殘差",
    "en": "Question 3797: Ordered Residual of a Specified Kernel Equation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r07-eigenvalue-0072",
    "batch_id": "batch-20261007-advance-20261009-r07-eigenvalue-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-eigenvalue"
    ],
    "title_zh": "指定核方程的有序殘差",
    "stem_zh": "實二階方陣與精確整數資料。M=[[22,1],[-1,-19]]，λ=-1，v=[1, 1]。列Mv−λv的兩個整數座標，保留順序。",
    "answer": {
      "kind": "expression",
      "value": "(24, -19)",
      "explanation_zh": "核殘差=(24, -19)。只是核方程殘差；零向量可有零殘差，不能因此稱作特徵向量。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "指定核方程的有序殘差"
      ]
    },
    "variant": {
      "family_id": "qf-density-eigen-kernel-residual",
      "seed": 202610097072,
      "parameters": {
        "M": [
          [
            22,
            1
          ],
          [
            -1,
            -19
          ]
        ],
        "lambda": -1,
        "v": [
          1,
          1
        ],
        "operation": "density_eigen_kernel_residual",
        "domain": "real_two_by_two",
        "semantic_template": "qf-density-eigen-kernel-residual/[[[22,1],[-1,-19]],-1,[1,1]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261009-r07-eigenvalue-0072",
    "batch_id": "batch-20261007-advance-20261009-r07-eigenvalue-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9692adc666eb1ce939ae1f366b786ed5312cd9eb504d6856e8e9668b10daa100",
      "family_id": "qf-density-eigen-kernel-residual",
      "seed": 202610097072,
      "semantic_template": "qf-density-eigen-kernel-residual/[[[22,1],[-1,-19]],-1,[1,1]]"
    },
    "fields": {
      "title": "Ordered Residual of a Specified Kernel Equation",
      "stem": "Over R, M=[[22, 1], [-1, -19]] is 2×2, λ=-1, and v=[1, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.",
      "answer": {
        "kind": "expression",
        "value": "(24, -19)",
        "explanation": "Exact multiplication and subtraction yield (24, -19). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."
      },
      "skills": [
        "Ordered Residual of a Specified Kernel Equation"
      ],
      "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": "實二階方陣與精確整數資料。M=[[22,1],[-1,-19]]，λ=-1，v=[1, 1]。列Mv−λv的兩個整數座標，保留順序。",
        "en": "Over R, M=[[22, 1], [-1, -19]] is 2×2, λ=-1, and v=[1, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "[[22, 1], [-1, -19]]",
      "after": "exact real2×2 matrix;no spectrum or diagonalization inferred",
      "relation": "implies",
      "rule_id": "two_by_two_matrix_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只處理宣告的實二階矩陣；係數、判別式或單一候選殘差都不自動給出完整譜或可對角化性。",
        "en": "Use the declared real2×2 matrix;coefficients,a discriminant or one candidate residual does not automatically establish the entire spectrum or diagonalizability."
      },
      "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": "row1 product1 := mul(22, 1)",
      "after": "row1 product1 = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩陣行與向量的對應分量相乘，不把行列索引交換。",
        "en": "Multiply corresponding row and vector components without interchanging row and column indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 row1 product1，不以小數近似替代。",
          "en": "Recompute row1 product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "1"
        ],
        "output": "22",
        "label": "row1 product1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "row1 product2 := mul(1, 1)",
      "after": "row1 product2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩陣行與向量的對應分量相乘，不把行列索引交換。",
        "en": "Multiply corresponding row and vector components without interchanging row and column indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 row1 product2，不以小數近似替代。",
          "en": "Recompute row1 product2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "row1 product2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Mv coordinate1 := sum(22, 1)",
      "after": "Mv coordinate1 = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一行的乘積相加得到Mv的一個分量。",
        "en": "Add the row products to obtain one component of Mv."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv coordinate1，不以小數近似替代。",
          "en": "Recompute Mv coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "1"
        ],
        "output": "23",
        "label": "Mv coordinate1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "λv coordinate1 := mul(-1, 1)",
      "after": "λv coordinate1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "標量λ乘同一索引向量分量。",
        "en": "Multiply λ by the vector component at the same index."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 λv coordinate1，不以小數近似替代。",
          "en": "Recompute λv coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "λv coordinate1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "residual coordinate1 := sub(23, -1)",
      "after": "residual coordinate1 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐座標取Mv−λv；零殘差才符合此方程。",
        "en": "Subtract λv from Mv coordinatewise;zero residual satisfies this equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 residual coordinate1，不以小數近似替代。",
          "en": "Recompute residual coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "23",
          "-1"
        ],
        "output": "24",
        "label": "residual coordinate1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "row2 product1 := mul(-1, 1)",
      "after": "row2 product1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩陣行與向量的對應分量相乘，不把行列索引交換。",
        "en": "Multiply corresponding row and vector components without interchanging row and column indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 row2 product1，不以小數近似替代。",
          "en": "Recompute row2 product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "row2 product1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "row2 product2 := mul(-19, 1)",
      "after": "row2 product2 = -19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩陣行與向量的對應分量相乘，不把行列索引交換。",
        "en": "Multiply corresponding row and vector components without interchanging row and column indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 row2 product2，不以小數近似替代。",
          "en": "Recompute row2 product2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-19",
          "1"
        ],
        "output": "-19",
        "label": "row2 product2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Mv coordinate2 := sum(-1, -19)",
      "after": "Mv coordinate2 = -20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一行的乘積相加得到Mv的一個分量。",
        "en": "Add the row products to obtain one component of Mv."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv coordinate2，不以小數近似替代。",
          "en": "Recompute Mv coordinate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "-19"
        ],
        "output": "-20",
        "label": "Mv coordinate2"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "λv coordinate2 := mul(-1, 1)",
      "after": "λv coordinate2 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "標量λ乘同一索引向量分量。",
        "en": "Multiply λ by the vector component at the same index."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 λv coordinate2，不以小數近似替代。",
          "en": "Recompute λv coordinate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "λv coordinate2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "residual coordinate2 := sub(-20, -1)",
      "after": "residual coordinate2 = -19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐座標取Mv−λv；零殘差才符合此方程。",
        "en": "Subtract λv from Mv coordinatewise;zero residual satisfies this equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 residual coordinate2，不以小數近似替代。",
          "en": "Recompute residual coordinate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-20",
          "-1"
        ],
        "output": "-19",
        "label": "residual coordinate2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "v=[1, 1];residual=(24, -19)",
      "after": "kernel equation is not an eigenvector certificate for v=0",
      "relation": "implies",
      "rule_id": "zero_vector_kernel_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零向量對任意λ都有零殘差；特徵向量另外要求非零，不能從核殘差0省略此前提。",
        "en": "The zero vector has zero residual for every λ;an eigenvector must additionally be nonzero,and a zero kernel residual cannot remove that premise."
      },
      "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": [
      "24",
      "-19"
    ],
    "residual": [
      "24",
      "-19"
    ]
  },
  "conclusion": {
    "canonical_value": "(24, -19)",
    "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-20261007-advance-20261009-r07-eigenvalue-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r07-eigenvalue-foundations/questions.jsonl",
      "batch_sha256": "e7cddab340faf1ec3ec8afd1de71da0f2940357248570ee9b85309e2cdfda2be",
      "original_line": 72,
      "original_line_sha256": "1b61c60ea67f80bd9a8152802331eb62126d75658b2202158c1159b1d1851881",
      "question_compact_sha256": "9692adc666eb1ce939ae1f366b786ed5312cd9eb504d6856e8e9668b10daa100",
      "english_row_compact_sha256": "5205a309d96131661ebee98bfbdfa7598bfae96698d893c1faa7aa3078743a59",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r07-eigenvalue-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r07-eigenvalue-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-eigenvalue": {
        "sha256": "5e1e1186120be48f1e86332bf1d679143e44074e2b3e9f59c5c219cb76a99e36",
        "english_sha256": "e48363ac57349147d31e8427dc2281ab02ae34f1abaea817c89e3e0f431fe5ed",
        "original_path": "public/data/mko/eigenvalue.json",
        "english_path": "public/data/i18n/editions/mko-eigenvalue.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6",
    "original_candidate_sha256": "5b8f6e21abcea5de9611adba117301e41aa8bd1c6635e7caa3cdeff8c2f415af",
    "original_candidate_line_sha256": "5b8f6e21abcea5de9611adba117301e41aa8bd1c6635e7caa3cdeff8c2f415af"
  },
  "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-20261026-004-p7",
    "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": "568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d",
      "question_first_witness_sha256": "d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 12,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "77368dfad6c08daa2736ae52e05db8603274a9003e7dfa035d70ccf6c7444bf5",
      "arithmetic_steps": 10
    },
    "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."
  }
}