{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r01-vector-0031",
  "pilot_number": "R0014-07931",
  "number": 7931,
  "family_id": "qf-density16-vector-difference-reversal",
  "title": {
    "zh-Hant": "第 7931 題：向量差的兩個順序",
    "en": "Question 7931: Both Directed Vector Differences and Their Sum"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r01-vector-0031",
    "batch_id": "batch-20261012-advance-20261016-r01-vector-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-vector"
    ],
    "title_zh": "向量差的兩個順序",
    "stem_zh": "固定標準基底的整數二分量向量。u=[-7, 1]、v=[0, -1]。依序答 (u, v, u−v, v−u, 兩個差的和)。",
    "answer": {
      "kind": "expression",
      "value": "([-7, 1], [0, -1], [-7, 2], [7, -2], [0, 0])",
      "explanation_zh": "有限結構核對=([-7, 1], [0, -1], [-7, 2], [7, -2], [0, 0])。保留有序二分量，兩種減法各算一次再相加。有限精確核對不是一般形式證明。逐分量不當內積或長度。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "向量差的兩個順序"
      ]
    },
    "variant": {
      "family_id": "qf-density16-vector-difference-reversal",
      "seed": 202610161031,
      "parameters": {
        "u": [
          -7,
          1
        ],
        "v": [
          0,
          -1
        ],
        "domain": "Z2_fixed_basis",
        "operation": "vector-difference-reversal",
        "semantic_template": "qf-density16-vector-difference-reversal/{\"u\":[-7,1],\"v\":[0,-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-r01-vector-0031",
    "batch_id": "batch-20261012-advance-20261016-r01-vector-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3ea4cf435a77c8c60f877fda3af77b92a7f4489be2ea5011ecd349a186894c5e",
      "family_id": "qf-density16-vector-difference-reversal",
      "seed": 202610161031,
      "semantic_template": "qf-density16-vector-difference-reversal/{\"u\":[-7,1],\"v\":[0,-1]}"
    },
    "fields": {
      "title": "Both Directed Vector Differences and Their Sum",
      "stem": "Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).",
      "answer": {
        "kind": "expression",
        "value": "([-7, 1], [0, -1], [-7, 2], [7, -2], [0, 0])",
        "explanation": "Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 1], [0, -1], [-7, 2], [7, -2], [0, 0])."
      },
      "skills": [
        "Both Directed Vector Differences and Their Sum"
      ],
      "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": "固定標準基底的整數二分量向量。u=[-7, 1]、v=[0, -1]。依序答 (u, v, u−v, v−u, 兩個差的和)。",
        "en": "Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "fixed standard basis;two ordered integer coordinates",
      "after": "componentwise operations;not length or inner product",
      "relation": "implies",
      "rule_id": "ordered_vector_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一固定基底下逐分量運算；交換分量、取長度或算內積都不是此題的向量答案。",
        "en": "Work coordinate by coordinate in the same fixed basis.Swapping coordinates,taking a norm or computing an inner product is not the requested vector answer."
      },
      "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": "u-v[0] := sub(-7, 0)",
      "after": "u-v[0] = -7",
      "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 u-v[0]，不以小數近似替代。",
          "en": "Recompute u-v[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-7",
          "0"
        ],
        "output": "-7",
        "label": "u-v[0]"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "u-v[1] := sub(1, -1)",
      "after": "u-v[1] = 2",
      "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 u-v[1]，不以小數近似替代。",
          "en": "Recompute u-v[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "2",
        "label": "u-v[1]"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "v-u[0] := sub(0, -7)",
      "after": "v-u[0] = 7",
      "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 v-u[0]，不以小數近似替代。",
          "en": "Recompute v-u[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-7"
        ],
        "output": "7",
        "label": "v-u[0]"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "v-u[1] := sub(-1, 1)",
      "after": "v-u[1] = -2",
      "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 v-u[1]，不以小數近似替代。",
          "en": "Recompute v-u[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-2",
        "label": "v-u[1]"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "opposite differences sum[0] := sum(-7, 7)",
      "after": "opposite differences sum[0] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 opposite differences sum[0]，不以小數近似替代。",
          "en": "Recompute opposite differences sum[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-7",
          "7"
        ],
        "output": "0",
        "label": "opposite differences sum[0]"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "opposite differences sum[1] := sum(2, -2)",
      "after": "opposite differences sum[1] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 opposite differences sum[1]，不以小數近似替代。",
          "en": "Recompute opposite differences sum[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "-2"
        ],
        "output": "0",
        "label": "opposite differences sum[1]"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "u=[-7, 1];v=[0, -1];u-v=[-7, 2];v-u=[7, -2]",
      "after": "sum=[0, 0]",
      "relation": "implies",
      "rule_id": "opposite_vector_difference_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩方向分別算減法，逐分量相加為零；u−v與v−u一般不相等，而是互為相反向量。",
        "en": "Compute both subtraction orders separately.Their componentwise sum is zero;u−v and v−u are generally opposites,not equal."
      },
      "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,
        1
      ],
      [
        0,
        -1
      ],
      [
        "-7",
        "2"
      ],
      [
        "7",
        "-2"
      ],
      [
        "0",
        "0"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-7, 1], [0, -1], [-7, 2], [7, -2], [0, 0])",
    "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-r01-vector-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r01-vector-foundations/questions.jsonl",
      "batch_sha256": "3bfc998f6a115fe8a36b505fbe6d5ba211ef040180b9b8fa8684df4ece6c7261",
      "original_line": 31,
      "original_line_sha256": "78256a9bd84a4475bcdf287844199f50d75771e32c9f51c78474cda1b03c29aa",
      "question_compact_sha256": "3ea4cf435a77c8c60f877fda3af77b92a7f4489be2ea5011ecd349a186894c5e",
      "english_row_compact_sha256": "3b5c97c87618a5f487f2e29387e1b9970937ad852fa25ac168463b23051a40cb",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r01-vector-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r01-vector-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-vector": {
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        "original_path": "public/data/mko/vector.json",
        "english_path": "public/data/i18n/editions/mko-vector.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d7860245c70062705fa4858c13821db6ff286678dec63f22464364f5d24cfcc1",
    "original_candidate_sha256": "34d686aac9004e2ac2b6d0cf79e5f66bb92512783d88cb7af01d6a62eb3441ba",
    "original_candidate_line_sha256": "34d686aac9004e2ac2b6d0cf79e5f66bb92512783d88cb7af01d6a62eb3441ba"
  },
  "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": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "e3e07387815b2c7d85ee1c5a797444033514ad6e54714186b95095cdec962e21",
      "arithmetic_steps": 6
    },
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    "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."
  }
}