{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r04-determinant-0092",
  "pilot_number": "R0013-07317",
  "number": 7317,
  "family_id": "qf-density16-det-candidate-absolute-audit",
  "title": {
    "zh-Hant": "第 7317 題：有號D與絕對值候選",
    "en": "Question 7317: Signed Determinant versus Absolute-Value Candidate"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r04-determinant-0092",
    "batch_id": "batch-20261012-advance-20261016-r04-determinant-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-determinant"
    ],
    "title_zh": "有號D與絕對值候選",
    "stem_zh": "R上2×2矩陣，D為有號行列式。M=[[4, 1], [2, 1]]，整數候選=2。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。",
    "answer": {
      "kind": "expression",
      "value": "([[4, 1], [2, 1]], 2, 2, 0, 是, 是)",
      "explanation_zh": "有限結構核對=([[4, 1], [2, 1]], 2, 2, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有號D與絕對值候選"
      ]
    },
    "variant": {
      "family_id": "qf-density16-det-candidate-absolute-audit",
      "seed": 202610164092,
      "parameters": {
        "M": [
          [
            4,
            1
          ],
          [
            2,
            1
          ]
        ],
        "candidate": 2,
        "domain": "R_2x2_integer_slice",
        "operation": "det-candidate-absolute-audit",
        "semantic_template": "qf-density16-det-candidate-absolute-audit/{\"M\":[[4,1],[2,1]],\"candidate\":2}"
      }
    },
    "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": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261016-r04-determinant-0092",
    "batch_id": "batch-20261012-advance-20261016-r04-determinant-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5e32cf786078342b648c55417dadfd078876744653f16f5bdf4adde1a030be70",
      "family_id": "qf-density16-det-candidate-absolute-audit",
      "seed": 202610164092,
      "semantic_template": "qf-density16-det-candidate-absolute-audit/{\"M\":[[4,1],[2,1]],\"candidate\":2}"
    },
    "fields": {
      "title": "Signed Determinant versus Absolute-Value Candidate",
      "stem": "Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [2, 1]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).",
      "answer": {
        "kind": "expression",
        "value": "([[4, 1], [2, 1]], 2, 2, 0, Yes, Yes)",
        "explanation": "Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 1], [2, 1]], 2, 2, 0, Yes, Yes)."
      },
      "skills": [
        "Signed Determinant versus Absolute-Value 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": "R上2×2矩陣，D為有號行列式。M=[[4, 1], [2, 1]]，整數候選=2。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。",
        "en": "Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [2, 1]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "[[4, 1], [2, 1]]",
      "after": "real2×2 integer-entry slice;determinants are signed",
      "relation": "implies",
      "rule_id": "signed_two_by_two_determinant_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行列式按有號ad−bc計算，不以絕對值取代；矩陣乘法按行乘列，不當逐項乘法。此有限核對不是一般形式證明。",
        "en": "Compute the signed determinant ad−bc,not its absolute value;matrix multiplication uses row-by-column,not entrywise products.This finite audit is not a general formal proof."
      },
      "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": "M diagonal product := mul(4, 1)",
      "after": "M diagonal product = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘主對角線a·d。",
        "en": "Multiply the main diagonal entries a·d."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M diagonal product，不以小數近似替代。",
          "en": "Recompute M diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1"
        ],
        "output": "4",
        "label": "M diagonal product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "M off-diagonal product := mul(1, 2)",
      "after": "M off-diagonal product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另乘副對角線b·c，減號不可漏。",
        "en": "Multiply the off-diagonal entries b·c separately,retaining subtraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M off-diagonal product，不以小數近似替代。",
          "en": "Recompute M off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "M off-diagonal product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "M signed determinant := sub(4, 2)",
      "after": "M signed determinant = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減保留符號。",
        "en": "Subtract the terms,retaining the sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M signed determinant，不以小數近似替代。",
          "en": "Recompute M signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "2"
        ],
        "output": "2",
        "label": "M signed determinant"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "candidate minus signed determinant := sub(2, 2)",
      "after": "candidate minus signed determinant = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選對照的是有號D，不是abs(D)。",
        "en": "Compare the candidate with signed D,not abs(D)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate minus signed determinant，不以小數近似替代。",
          "en": "Recompute candidate minus signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "candidate minus signed determinant"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "signed D=2;abs(D)=2;candidate=2;defect=0",
      "after": "candidate correct=是;invertible over R=是",
      "relation": "implies",
      "rule_id": "signed_candidate_vs_invertibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選正確與可逆是兩個資格；D=0的正確候選也不代表可逆，D<0時絕對值候選通常不等於D。",
        "en": "Candidate correctness and invertibility are separate:a correct candidate D=0 does not make the matrix invertible,and for D<0 an absolute-value candidate differs from D."
      },
      "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": [
      [
        [
          4,
          1
        ],
        [
          2,
          1
        ]
      ],
      "2",
      "2",
      "0",
      true,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([[4, 1], [2, 1]], 2, 2, 0, 是, 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-20261012-advance-20261016-r04-determinant-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r04-determinant-foundations/questions.jsonl",
      "batch_sha256": "e8a897b13857e0c229dad6a8865c2007e594f9c51f688f9113b353ff2562dd0a",
      "original_line": 92,
      "original_line_sha256": "bb971baf5de8f0c06dcd6a8d27b0f6a7045abdc9a5774cff77b675f61286de62",
      "question_compact_sha256": "5e32cf786078342b648c55417dadfd078876744653f16f5bdf4adde1a030be70",
      "english_row_compact_sha256": "100cd40836c6f7bd6d9b860956b8eb093555ea93a3411d31823646f54a1cbc42",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r04-determinant-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r04-determinant-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-determinant": {
        "sha256": "d49050d70b58e0d9d2cab2f5ddc64133813f5a99418865b5941af4f1ec240093",
        "english_sha256": "c20e33f6fef226f483add30aa030f5fa68b5878d5ea16a6e3e75b0b360299311",
        "original_path": "public/data/mko/determinant.json",
        "english_path": "public/data/i18n/editions/mko-determinant.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "944e5a6e7ddc77eb28f9280c1b90da6c7083069b4790cbeb7488d009328efce8",
    "original_candidate_sha256": "4e5089785ddaf694122dc55eb1cc2c899996a8e8c330e50bfad81414ebed171d",
    "original_candidate_line_sha256": "4e5089785ddaf694122dc55eb1cc2c899996a8e8c330e50bfad81414ebed171d"
  },
  "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-20261026-005-p14",
    "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": "e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59",
      "question_first_witness_sha256": "0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2758f0436c702171333df007107796ca70d961539f174d5fcc3e2e654b64e448",
      "arithmetic_steps": 4
    },
    "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."
  }
}