{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261006-advance-20261007-r04-eigenvalue-0055",
  "pilot_number": "R0061-31480",
  "number": 31480,
  "family_id": "qf-eigenvalue-kernel-membership",
  "title": {
    "zh-Hant": "第 31480 題：候選純量的 kernel 成員",
    "en": "Question 31480: Kernel Membership for a Candidate Scalar"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261006-advance-20261007-r04-eigenvalue-0055",
    "batch_id": "batch-20261006-advance-20261007-r04-eigenvalue-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-eigenvalue"
    ],
    "title_zh": "候選純量的 kernel 成員",
    "stem_zh": "實二階方陣，論域 R，M=[[4, 1], [1, 4]]。λ=5、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。",
    "answer": {
      "kind": "short_text",
      "value": "是",
      "explanation_zh": "核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "候選純量的 kernel 成員"
      ]
    },
    "variant": {
      "family_id": "qf-eigenvalue-kernel-membership",
      "seed": 202610074055,
      "parameters": {
        "M": [
          [
            4,
            1
          ],
          [
            1,
            4
          ]
        ],
        "lambda": 5,
        "v": [
          1,
          1
        ],
        "operation": "kernel_membership",
        "domain": "reals",
        "semantic_template": "qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"4\",\"1\"],[\"1\",\"4\"]],\"5\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"
      }
    },
    "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": "2026-10-06",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261006-advance-20261007-r04-eigenvalue-0055",
    "batch_id": "batch-20261006-advance-20261007-r04-eigenvalue-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "922a79a98710cda4b1717df504aea37a3b4ab276790ebece84e47a00561f9ca2",
      "family_id": "qf-eigenvalue-kernel-membership",
      "seed": 202610074055,
      "semantic_template": "qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"4\",\"1\"],[\"1\",\"4\"]],\"5\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"
    },
    "fields": {
      "title": "Kernel Membership for a Candidate Scalar",
      "stem": "Over R, the 2×2 matrix is M=[[4, 1], [1, 4]]. For the candidate scalar λ=5 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."
      },
      "skills": [
        "Kernel Membership for a Candidate Scalar"
      ],
      "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，論域 R，M=[[4, 1], [1, 4]]。λ=5、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。",
        "en": "Over R, the 2×2 matrix is M=[[4, 1], [1, 4]]. For the candidate scalar λ=5 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-eigenvalue-kernel-membership, kernel_membership, domain, reals]",
      "after": "derived = [[equality_including_zero_vector_YesNo]]",
      "relation": "implies",
      "rule_id": "R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核本題欄位與來源協議：distance只用完整座標frame，其他用domain；離散運算local-hyphen，舊bank用underscore。輸出欄序及型別依題族，保留原始中英資料與狀態。",
        "en": "Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Eigen ORIGINAL row0 term0 := mul(4, 1)",
      "after": "Eigen ORIGINAL row0 term0 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL row0 term0，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL row0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1"
        ],
        "output": "4",
        "label": "Eigen ORIGINAL row0 term0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Eigen ORIGINAL row0 term1 := mul(1, 1)",
      "after": "Eigen ORIGINAL row0 term1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL row0 term1，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL row0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Eigen ORIGINAL row0 term1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Eigen ORIGINAL Mv row0 sum := sum(4, 1)",
      "after": "Eigen ORIGINAL Mv row0 sum = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL Mv row0 sum，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL Mv row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4",
          "1"
        ],
        "output": "5",
        "label": "Eigen ORIGINAL Mv row0 sum"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Eigen ORIGINAL lambda times v row0 := mul(5, 1)",
      "after": "Eigen ORIGINAL lambda times v row0 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL lambda times v row0，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL lambda times v row0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "1"
        ],
        "output": "5",
        "label": "Eigen ORIGINAL lambda times v row0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Eigen ORIGINAL residual row0 := sub(5, 5)",
      "after": "Eigen ORIGINAL residual row0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL residual row0，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL residual row0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "5"
        ],
        "output": "0",
        "label": "Eigen ORIGINAL residual row0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Eigen ORIGINAL row1 term0 := mul(1, 1)",
      "after": "Eigen ORIGINAL row1 term0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL row1 term0，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL row1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Eigen ORIGINAL row1 term0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Eigen ORIGINAL row1 term1 := mul(4, 1)",
      "after": "Eigen ORIGINAL row1 term1 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL row1 term1，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL row1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1"
        ],
        "output": "4",
        "label": "Eigen ORIGINAL row1 term1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Eigen ORIGINAL Mv row1 sum := sum(1, 4)",
      "after": "Eigen ORIGINAL Mv row1 sum = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL Mv row1 sum，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL Mv row1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "4"
        ],
        "output": "5",
        "label": "Eigen ORIGINAL Mv row1 sum"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Eigen ORIGINAL lambda times v row1 := mul(5, 1)",
      "after": "Eigen ORIGINAL lambda times v row1 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL lambda times v row1，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL lambda times v row1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "1"
        ],
        "output": "5",
        "label": "Eigen ORIGINAL lambda times v row1"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Eigen ORIGINAL residual row1 := sub(5, 5)",
      "after": "Eigen ORIGINAL residual row1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Eigen ORIGINAL residual row1，不以小數近似替代。",
          "en": "Recompute Eigen ORIGINAL residual row1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "5"
        ],
        "output": "0",
        "label": "Eigen ORIGINAL residual row1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [[[4, 1], [1, 4]], 5, [1, 1], kernel_membership]",
      "after": "derived = [[5, 5], [5, 5], [0, 0], 是, 是, 是]",
      "relation": "implies",
      "rule_id": "Eigen_ALL_rows_equation_AND_NONZERO_eigenvector_vs_kernel_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐row驗Mv=λv，再分是否非零vector。eigenpair需要v≠0且等式真；kernel只需(M−λI)v=0，因此零向量永遠在kernel，不能據此說λ必為特徵值。",
        "en": "Check Mv=λv in EVERY row,then SEPARATELY test nonzero v.An eigenpair needs BOTH;kernel membership needs only(M−λI)v=0,so the zero vector is always in the kernel WITHOUT establishing that λ is an eigenvalue."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "是"
  },
  "conclusion": {
    "canonical_value": "是",
    "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-20261006-advance-20261007-r04-eigenvalue-foundations",
      "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r04-eigenvalue-foundations/questions.jsonl",
      "batch_sha256": "d81d816856425974bbc3c6986818908fe3b9c0210bc8ab9f4c988e781227b637",
      "original_line": 55,
      "original_line_sha256": "742749d8e5acf02b323732e64de118dc521f0a593265cf80b06ed83acbd6dae8",
      "question_compact_sha256": "922a79a98710cda4b1717df504aea37a3b4ab276790ebece84e47a00561f9ca2",
      "english_row_compact_sha256": "c77c2682b43c27f6752776d36617db01468717c5fbe928cffdf65bf32a8e786e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r04-eigenvalue-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r04-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": "912db26eb6c90aa6e256f4356cb5ee94cd0a352f04f667876ae6739f8181a17c",
    "original_candidate_sha256": "b459df7d829463bd3736d015c8b70aa2d2a2c87805486f36806e6a2812d5807c",
    "original_candidate_line_sha256": "b459df7d829463bd3736d015c8b70aa2d2a2c87805486f36806e6a2812d5807c"
  },
  "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-20261026-011-p62",
    "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": "5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69",
      "question_first_witness_sha256": "c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495"
    },
    "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": "04d3f7b1b53a2175e0fd8707acf3c0dbdc2b84c660bca89bf8f716e25f5ad050",
      "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."
  }
}