{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261006-advance-20261007-r07-geometric-transformations-0097",
  "pilot_number": "R0058-29797",
  "number": 29797,
  "family_id": "qf-transform-uniform-scale-class",
  "title": {
    "zh-Hant": "第 29797 題：全平面均勻縮放分類",
    "en": "Question 29797: Whole-Plane Classification of Uniform Scaling"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261006-advance-20261007-r07-geometric-transformations-0097",
    "batch_id": "batch-20261006-advance-20261007-r07-geometric-transformations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-geometric-transformations"
    ],
    "title_zh": "全平面均勻縮放分類",
    "stem_zh": "等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-21P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。",
    "answer": {
      "kind": "short_text",
      "value": "非等距相似映射；倍率=21",
      "explanation_zh": "全域係數=-21；長度倍率=21。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "全平面均勻縮放分類"
      ]
    },
    "variant": {
      "family_id": "qf-transform-uniform-scale-class",
      "seed": 202610077097,
      "parameters": {
        "c": -21,
        "operation": "uniform_scale_class",
        "frame": "euclidean_orthonormal_equal_units",
        "semantic_template": "qf-transform-uniform-scale-class/{\"inputs\":[\"21\"],\"operation\":\"uniform_scale_class\"}"
      }
    },
    "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-r07-geometric-transformations-0097",
    "batch_id": "batch-20261006-advance-20261007-r07-geometric-transformations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3e2841785e732314a3c93d4b9caf26f9ada3fbf176fd9521719f1b77d259e049",
      "family_id": "qf-transform-uniform-scale-class",
      "seed": 202610077097,
      "semantic_template": "qf-transform-uniform-scale-class/{\"inputs\":[\"21\"],\"operation\":\"uniform_scale_class\"}"
    },
    "fields": {
      "title": "Whole-Plane Classification of Uniform Scaling",
      "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-21)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.",
      "answer": {
        "kind": "short_text",
        "value": "non-isometric similarity; scale=21",
        "explanation": "Classify the global formula by |c|, giving non-isometric similarity; scale=21: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."
      },
      "skills": [
        "Whole-Plane Classification of Uniform Scaling"
      ],
      "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": "等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-21P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。",
        "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-21)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-transform-uniform-scale-class, uniform_scale_class]",
      "after": "derived = [[WHOLEplane_map_class_label_positive_length_scale]]",
      "relation": "implies",
      "rule_id": "R58_EXACT_source_interface_and_answer_ROLE",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本題銀行的keys、domain或frame、operation與輸出角色。STR22/App20 tuple與legacy scalar不同；分數是否保/1按原契約，不能用共同外觀改題。",
        "en": "Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。",
          "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [-21, euclidean_orthonormal_equal_units]",
      "after": "derived = [21, ||cP-cQ||=|c|*||P-Q||, 非等距相似映射；倍率=21]",
      "relation": "implies",
      "rule_id": "Transform_WHOLE_plane_distance_scaling_identity_ALL_P_Q",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對任意P,Q，向量差乘c，平方長乘c²，非負平方根使距離乘|c|。c0把全平面送原點；|c|1保全部距離；其他非零有正倍率|c|但非等距（取不同點即可）。一個重合pair不證global property；c=-1是原點180度旋轉，不是line reflection。",
        "en": "For ANY P,Q,the difference scales by c,its squared norm by c²,and the distance by|c|.At0 the whole plane collapses to the origin;|c|=1 preserves ALL distances;other nonzero c gives positive scale|c| but not isometry for distinct points.One coincident pair is not a global certificate;c=-1 is an origin180-degree rotation,not a line reflection."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。",
          "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "非等距相似映射；倍率=21"
  },
  "conclusion": {
    "canonical_value": "非等距相似映射；倍率=21",
    "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-r07-geometric-transformations-foundations",
      "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r07-geometric-transformations-foundations/questions.jsonl",
      "batch_sha256": "ba809546e64880b47dffa9a5e44ba757e9a0ce452af4b03745db86f595cb6b57",
      "original_line": 97,
      "original_line_sha256": "72c9ffbb7bb4997200548b122940a33c7ca322872cdc11b0913826dd0a7ff441",
      "question_compact_sha256": "3e2841785e732314a3c93d4b9caf26f9ada3fbf176fd9521719f1b77d259e049",
      "english_row_compact_sha256": "63cb239ba95c93bfd6b5266227c176954e9f917bf82ede3024038fee35236993",
      "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r07-geometric-transformations-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r07-geometric-transformations-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-geometric-transformations": {
        "sha256": "13176ddfbd967892818086ef2859a68ada71d09c120c932f688634fbec64e489",
        "english_sha256": "403883e15b18bf9b23ff01fe8eec209041ce5774af2f19c72ec39d5e53538c31",
        "original_path": "public/data/mko/geometric-transformations.json",
        "english_path": "public/data/i18n/editions/mko-geometric-transformations.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "623c23f45586b08629c5a347ffbc994cf8dbff2133c4f4cdb227b851d74ea6b6",
    "original_candidate_sha256": "afc9a311f59fa01240d4419acbf312b4b0c8a3007b73fcfadf970b2a96c792fa",
    "original_candidate_line_sha256": "afc9a311f59fa01240d4419acbf312b4b0c8a3007b73fcfadf970b2a96c792fa"
  },
  "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-p59",
    "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": "66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df",
      "question_first_witness_sha256": "b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 2,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "220b7748eeed7d149b33333603b8d478f8706219ae7f26451910943d6227a156",
      "arithmetic_steps": 0
    },
    "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."
  }
}