{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r02-matrix-0081",
  "pilot_number": "R0014-07606",
  "number": 7606,
  "family_id": "qf-density16-matrix-candidate-transpose-audit",
  "title": {
    "zh-Hant": "第 7606 題：轉置候選的尺寸與殘差",
    "en": "Question 7606: Candidate Transpose Shape before Entrywise Residuals"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r02-matrix-0081",
    "batch_id": "batch-20261012-advance-20261016-r02-matrix-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-matrix"
    ],
    "title_zh": "轉置候選的尺寸與殘差",
    "stem_zh": "外部索引從1開始；每格完整整數資料。M=[[-7, 1], [2, -1]]，候選矩陣=[[-6, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。",
    "answer": {
      "kind": "expression",
      "value": "([[-7, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)",
      "explanation_zh": "有限結構核對=([[-7, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 否)。先算轉置尺寸；候選同尺寸才逐格相減，尺寸不合時不補造殘差。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "轉置候選的尺寸與殘差"
      ]
    },
    "variant": {
      "family_id": "qf-density16-matrix-candidate-transpose-audit",
      "seed": 202610162081,
      "parameters": {
        "M": [
          [
            -7,
            1
          ],
          [
            2,
            -1
          ]
        ],
        "candidate": [
          [
            -6,
            2
          ],
          [
            1,
            -1
          ]
        ],
        "domain": "Z_rect_1based",
        "operation": "matrix-candidate-transpose-audit",
        "semantic_template": "qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-7,1],[2,-1]],\"candidate\":[[-6,2],[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": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261016-r02-matrix-0081",
    "batch_id": "batch-20261012-advance-20261016-r02-matrix-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "351a6fbd3acebdcf7a6edd8b5566509474271e993728e2a800e6cbab491461d9",
      "family_id": "qf-density16-matrix-candidate-transpose-audit",
      "seed": 202610162081,
      "semantic_template": "qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-7,1],[2,-1]],\"candidate\":[[-6,2],[1,-1]]}"
    },
    "fields": {
      "title": "Candidate Transpose Shape before Entrywise Residuals",
      "stem": "Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-7, 1], [2, -1]], candidate=[[-6, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).",
      "answer": {
        "kind": "expression",
        "value": "([[-7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)",
        "explanation": "Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."
      },
      "skills": [
        "Candidate Transpose Shape before Entrywise Residuals"
      ],
      "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": "外部索引從1開始；每格完整整數資料。M=[[-7, 1], [2, -1]]，候選矩陣=[[-6, 2], [1, -1]]。求 (Mᵀ, 候選尺寸, 尺寸資格, 候選減Mᵀ的殘差矩陣或尺寸不合, 候選相等資格)。",
        "en": "Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-7, 1], [2, -1]], candidate=[[-6, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "M=[[-7, 1], [2, -1]];shape=[2, 2]",
      "after": "external indices start at1;transpose is not inverse",
      "relation": "implies",
      "rule_id": "rectangular_matrix_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先確認非空矩形尺寸與完整整數資料。轉置對矩形矩陣也可定義，不要求可逆，更不是求逆。",
        "en": "First confirm a nonempty rectangular shape and complete integer entries.Transposition is defined for rectangular matrices without invertibility and is not matrix inversion."
      },
      "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 transpose:source(1,1)=-7",
      "after": "target(1,1)=-7",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "M transpose:source(1,2)=1",
      "after": "target(2,1)=1",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "M transpose:source(2,1)=2",
      "after": "target(1,2)=2",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M transpose:source(2,2)=-1",
      "after": "target(2,2)=-1",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "candidate shape=[2, 2];required transpose shape=[2, 2]",
      "after": "qualified=是",
      "relation": "implies",
      "rule_id": "candidate_matrix_shape_before_difference",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原M的行列數交換後才是候選必須符合的尺寸；尺寸不合，不建立逐格殘差。",
        "en": "The candidate must have the swapped row/column sizes of M.A shape mismatch forbids constructing an entrywise residual."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "candidate-MT(1,1) := sub(-6, -7)",
      "after": "candidate-MT(1,1) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按題目指定的前項減後項，保留殘差的正負。",
        "en": "Subtract the second operand from the first in the requested order,retaining the residual sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate-MT(1,1)，不以小數近似替代。",
          "en": "Recompute candidate-MT(1,1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-6",
          "-7"
        ],
        "output": "1",
        "label": "candidate-MT(1,1)"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "candidate-MT(1,2) := sub(2, 2)",
      "after": "candidate-MT(1,2) = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按題目指定的前項減後項，保留殘差的正負。",
        "en": "Subtract the second operand from the first in the requested order,retaining the residual sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate-MT(1,2)，不以小數近似替代。",
          "en": "Recompute candidate-MT(1,2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "candidate-MT(1,2)"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "candidate-MT(2,1) := sub(1, 1)",
      "after": "candidate-MT(2,1) = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按題目指定的前項減後項，保留殘差的正負。",
        "en": "Subtract the second operand from the first in the requested order,retaining the residual sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate-MT(2,1)，不以小數近似替代。",
          "en": "Recompute candidate-MT(2,1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1"
        ],
        "output": "0",
        "label": "candidate-MT(2,1)"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "candidate-MT(2,2) := sub(-1, -1)",
      "after": "candidate-MT(2,2) = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按題目指定的前項減後項，保留殘差的正負。",
        "en": "Subtract the second operand from the first in the requested order,retaining the residual sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate-MT(2,2)，不以小數近似替代。",
          "en": "Recompute candidate-MT(2,2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "0",
        "label": "candidate-MT(2,2)"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "qualified=是;residual/status=[[1, 0], [0, 0]]",
      "after": "equal=否",
      "relation": "implies",
      "rule_id": "candidate_matrix_all_entries",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合法尺寸下每一格殘差都0才相等；尺寸不合是先決條件失敗，不虛構零矩陣殘差。",
        "en": "With matching shapes every entry residual must be zero.Mismatched shapes fail a prerequisite and must not be assigned a fabricated zero residual matrix."
      },
      "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": [
      [
        [
          -7,
          2
        ],
        [
          1,
          -1
        ]
      ],
      [
        2,
        2
      ],
      true,
      [
        [
          "1",
          "0"
        ],
        [
          "0",
          "0"
        ]
      ],
      false
    ]
  },
  "conclusion": {
    "canonical_value": "([[-7, 2], [1, -1]], [2, 2], 是, [[1, 0], [0, 0]], 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-20261012-advance-20261016-r02-matrix-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r02-matrix-foundations/questions.jsonl",
      "batch_sha256": "2cfe67df8364350332cadfafbf7125a57390570620e1132ca9237e218f8c4cf8",
      "original_line": 81,
      "original_line_sha256": "5df5e67d48326550c667fd5114e613df957fcaa4358107423ba53f605a655210",
      "question_compact_sha256": "351a6fbd3acebdcf7a6edd8b5566509474271e993728e2a800e6cbab491461d9",
      "english_row_compact_sha256": "e88caadf1610667d7ad6c409b38a36cd2331cf76669bab5c4564cdae61229798",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r02-matrix-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r02-matrix-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-matrix": {
        "sha256": "c0233ecc6ee07427af1589feb16108500eb2e48b0b2fb84b5b39f0f38b02f109",
        "english_sha256": "18504021d52173d8f2f8aa663d83dc058f5205bea2c677a9722daba770776432",
        "original_path": "public/data/mko/matrix.json",
        "english_path": "public/data/i18n/editions/mko-matrix.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d7860245c70062705fa4858c13821db6ff286678dec63f22464364f5d24cfcc1",
    "original_candidate_sha256": "45c44c0d099a188392efd30a363c16becaec2ddbf4e5f5c479db0b78fc76ff90",
    "original_candidate_line_sha256": "45c44c0d099a188392efd30a363c16becaec2ddbf4e5f5c479db0b78fc76ff90"
  },
  "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-p15",
    "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": "23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77",
      "question_first_witness_sha256": "17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705"
    },
    "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": "a6040893a31634be21b1d29769f4212ee154e701cfb4e9e65873a5fae1cd3beb",
      "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."
  }
}