{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r07-eigenvalue-0099",
  "pilot_number": "R0006-03774",
  "number": 3774,
  "family_id": "qf-density-eigen-first-mismatch",
  "title": {
    "zh-Hant": "第 3774 題：非零向量候選的第一失配",
    "en": "Question 3774: First Mismatch of a Nonzero Eigenvector Candidate"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r07-eigenvalue-0099",
    "batch_id": "batch-20261007-advance-20261009-r07-eigenvalue-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-eigenvalue"
    ],
    "title_zh": "非零向量候選的第一失配",
    "stem_zh": "實二階方陣與精確整數資料。M=[[24,0],[0,50]]，λ=24，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。",
    "answer": {
      "kind": "short_text",
      "value": "2",
      "explanation_zh": "非零向量殘差=(0, 26)；第一失配=2。先核非零向量，再逐座標檢查；無失配只支持指定候選λ，不宣稱完整譜或對角化。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "非零向量候選的第一失配"
      ]
    },
    "variant": {
      "family_id": "qf-density-eigen-first-mismatch",
      "seed": 202610097099,
      "parameters": {
        "M": [
          [
            24,
            0
          ],
          [
            0,
            50
          ]
        ],
        "lambda": 24,
        "v": [
          1,
          1
        ],
        "operation": "density_eigen_first_mismatch",
        "domain": "real_two_by_two",
        "semantic_template": "qf-density-eigen-first-mismatch/[[[24,0],[0,50]],24,[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-0099",
    "batch_id": "batch-20261007-advance-20261009-r07-eigenvalue-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "835ee2a84638cf94c53344a533db53433b9e21d85681d9d09354c66a081d2386",
      "family_id": "qf-density-eigen-first-mismatch",
      "seed": 202610097099,
      "semantic_template": "qf-density-eigen-first-mismatch/[[[24,0],[0,50]],24,[1,1]]"
    },
    "fields": {
      "title": "First Mismatch of a Nonzero Eigenvector Candidate",
      "stem": "Over R, M=[[24, 0], [0, 50]] is 2×2, λ=24, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.",
      "answer": {
        "kind": "short_text",
        "value": "2",
        "explanation": "The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."
      },
      "skills": [
        "First Mismatch of a Nonzero Eigenvector Candidate"
      ],
      "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=[[24,0],[0,50]]，λ=24，非零v=[1, 1]。按座標1、2尋找Mv−λv第一個非零位置，皆零則答「無」。",
        "en": "Over R, M=[[24, 0], [0, 50]] is 2×2, λ=24, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "[[24, 0], [0, 50]]",
      "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(24, 1)",
      "after": "row1 product1 = 24",
      "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": [
          "24",
          "1"
        ],
        "output": "24",
        "label": "row1 product1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "row1 product2 := mul(0, 1)",
      "after": "row1 product2 = 0",
      "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": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "row1 product2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Mv coordinate1 := sum(24, 0)",
      "after": "Mv coordinate1 = 24",
      "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": [
          "24",
          "0"
        ],
        "output": "24",
        "label": "Mv coordinate1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "λv coordinate1 := mul(24, 1)",
      "after": "λv coordinate1 = 24",
      "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": [
          "24",
          "1"
        ],
        "output": "24",
        "label": "λv coordinate1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "residual coordinate1 := sub(24, 24)",
      "after": "residual coordinate1 = 0",
      "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": [
          "24",
          "24"
        ],
        "output": "0",
        "label": "residual coordinate1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "row2 product1 := mul(0, 1)",
      "after": "row2 product1 = 0",
      "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": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "row2 product1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "row2 product2 := mul(50, 1)",
      "after": "row2 product2 = 50",
      "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": [
          "50",
          "1"
        ],
        "output": "50",
        "label": "row2 product2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Mv coordinate2 := sum(0, 50)",
      "after": "Mv coordinate2 = 50",
      "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": [
          "0",
          "50"
        ],
        "output": "50",
        "label": "Mv coordinate2"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "λv coordinate2 := mul(24, 1)",
      "after": "λv coordinate2 = 24",
      "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": [
          "24",
          "1"
        ],
        "output": "24",
        "label": "λv coordinate2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "residual coordinate2 := sub(50, 24)",
      "after": "residual coordinate2 = 26",
      "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": [
          "50",
          "24"
        ],
        "output": "26",
        "label": "residual coordinate2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "[0, 26]",
      "after": "first nonzero coordinate=2",
      "relation": "implies",
      "rule_id": "nonzero_candidate_first_mismatch",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按座標1、2次序尋找首個失配；無失配且v非零只核可這個λ與v，不補做完整特徵值分類。",
        "en": "Find the first mismatch in coordinate order1,2;no mismatch with nonzero v validates only this λ,v pair,not a full eigenvalue classification."
      },
      "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": "2",
    "residual": [
      "0",
      "26"
    ]
  },
  "conclusion": {
    "canonical_value": "2",
    "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": 99,
      "original_line_sha256": "7eac824e32d924f1cdb897674182f2ba2052164128fd7dfdf716d30e756d9863",
      "question_compact_sha256": "835ee2a84638cf94c53344a533db53433b9e21d85681d9d09354c66a081d2386",
      "english_row_compact_sha256": "664ff978342f4601b36672d6681fc8b9f6250d71696c40a84376dcdbf394c475",
      "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": "522cf252304b379fd16984e01c57ce90981413b96fa5c10cfbbdd39a8efad520",
    "original_candidate_line_sha256": "522cf252304b379fd16984e01c57ce90981413b96fa5c10cfbbdd39a8efad520"
  },
  "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": "10b98abaaa2ed212e563acf350d630c13de907b0d69cf7a2bb30d67d41a7f218",
      "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."
  }
}