{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0001","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dcc373530d4525092e59996aaf23db31c2c9e8597cb441274e599faa63f87b0","family_id":"qf-distance-squared","seed":202610064001,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"0\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-12, 5) and Q=(-12, 5). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"0","explanation":"Add the squares of the two coordinate differences to obtain 0. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0002","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e470c6341bbe369be5e4894cf468cb0cb796ae5c7bfc7e30db7faa09bd3bf487","family_id":"qf-distance-squared","seed":202610064002,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"1\",\"2\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-11, 4) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"5","explanation":"Add the squares of the two coordinate differences to obtain 5. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0003","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76be72b561938da8217b162fa3f81e8ba767a4817fe26ed4deb9f7f7675d5b8e","family_id":"qf-distance-squared","seed":202610064003,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"2\",\"3\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-10, 3) and Q=(-8, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"13","explanation":"Add the squares of the two coordinate differences to obtain 13. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0004","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0e217ca1636923cda2ef601edd6a14bf5e8a705e47d8b4949ccb8f77a015853","family_id":"qf-distance-squared","seed":202610064004,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"3\",\"4\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-9, 2) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"25","explanation":"Add the squares of the two coordinate differences to obtain 25. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0005","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d686c1bc6eaa2668348880301853fbf1317d34a74244bc968a9d0fcbc7c6125e","family_id":"qf-distance-squared","seed":202610064005,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"4\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 5). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"16","explanation":"Add the squares of the two coordinate differences to obtain 16. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0006","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2180a0d03853a3b3d27a0007587c68056ca268a695d321bbaf8b4fcbf218884","family_id":"qf-distance-squared","seed":202610064006,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"5\",\"6\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"61","explanation":"Add the squares of the two coordinate differences to obtain 61. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0007","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b9e10e465bbde26f417aa0f89dcc8b88c8b99ebb422a99e6ee6c56e5bc3aa0d","family_id":"qf-distance-squared","seed":202610064007,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"6\",\"7\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(0, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"85","explanation":"Add the squares of the two coordinate differences to obtain 85. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0008","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25ea8ff9cde1f7463909009eef9e0b71d778ba47a1d6f9e46d53b61756d19906","family_id":"qf-distance-squared","seed":202610064008,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"7\",\"8\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-5, -2) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"113","explanation":"Add the squares of the two coordinate differences to obtain 113. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0009","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52405a2b0e89d1191799e54c23f2835f836c55f94a9da3063437b30e931d4026","family_id":"qf-distance-squared","seed":202610064009,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"8\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-4, -3) and Q=(-4, 5). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"64","explanation":"Add the squares of the two coordinate differences to obtain 64. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0010","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"090f36688cb2546dae69c97c3b2194e1bd658614a678cdfa41b1258cb7b61542","family_id":"qf-distance-squared","seed":202610064010,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"9\",\"10\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"181","explanation":"Add the squares of the two coordinate differences to obtain 181. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0011","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb9859eb993fe7dfc2998f210800ad2b6a71f9ab37b87c7d698aad488bc288c4","family_id":"qf-distance-squared","seed":202610064011,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"10\",\"11\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-2, -5) and Q=(8, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"221","explanation":"Add the squares of the two coordinate differences to obtain 221. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0012","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24638928022c237c7dcd2427fdeac3cdb1af65883cb40ad6ecef2e4409b7b9e3","family_id":"qf-distance-squared","seed":202610064012,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"11\",\"12\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-1, -6) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"265","explanation":"Add the squares of the two coordinate differences to obtain 265. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0013","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51c5b389c1b8aa379385fa0dd93f4084cbbc5bb896ff3a599d4b566c563c7208","family_id":"qf-distance-squared","seed":202610064013,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"12\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(0, -7) and Q=(0, 5). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"144","explanation":"Add the squares of the two coordinate differences to obtain 144. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0014","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f16eaeb67fb70731fe9576f9b03747169fe0d0bb26f01a5eb9c120d9095339f3","family_id":"qf-distance-squared","seed":202610064014,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"13\",\"14\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"365","explanation":"Add the squares of the two coordinate differences to obtain 365. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0015","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5bc508c8b7d4e869c4e7396f3fd556d68a750aa04cfe96e61300fa0858327dd","family_id":"qf-distance-squared","seed":202610064015,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"14\",\"15\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(2, -9) and Q=(16, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"421","explanation":"Add the squares of the two coordinate differences to obtain 421. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0016","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"497e85e161d95e853f9bebe787b811e160c0b037ca05bb644e3885138706b43e","family_id":"qf-distance-squared","seed":202610064016,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"15\",\"16\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"481","explanation":"Add the squares of the two coordinate differences to obtain 481. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0017","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6488bf9cc912825668f3f5c172717bb2e43faee7db7525a221727eefb236582","family_id":"qf-distance-squared","seed":202610064017,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"16\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 5). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"256","explanation":"Add the squares of the two coordinate differences to obtain 256. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0018","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3f704e749fa0f69b436a08c14c0b819dd30a8e900ed5248101bc76a6e142ce0","family_id":"qf-distance-squared","seed":202610064018,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"17\",\"18\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(5, -12) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"613","explanation":"Add the squares of the two coordinate differences to obtain 613. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0019","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71d421e7d540107116e5a339a0623ed8ce4818817704e0f3e20f1b2bada0d97b","family_id":"qf-distance-squared","seed":202610064019,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"18\",\"19\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(6, -13) and Q=(24, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"685","explanation":"Add the squares of the two coordinate differences to obtain 685. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0020","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"966fdc6d0c3bcdaaffedac44e0098a0ef1be9c1b36afc4b53b3016f92e60fbff","family_id":"qf-distance-squared","seed":202610064020,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"19\",\"20\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"761","explanation":"Add the squares of the two coordinate differences to obtain 761. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0021","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ea60c32116c38c5aeff25b619c0201021e8930a57d65be95281b53674e88122","family_id":"qf-distance-squared","seed":202610064021,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"20\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(8, -15) and Q=(8, 5). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"400","explanation":"Add the squares of the two coordinate differences to obtain 400. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0022","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef7c9eb1572f59dcbbb95969e42c9409df53407970fe48ff5913a4e577f5a548","family_id":"qf-distance-squared","seed":202610064022,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"21\",\"22\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(9, -16) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"925","explanation":"Add the squares of the two coordinate differences to obtain 925. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0023","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6481456da6816730c4aff22d801442243b0f801e9b6965cd542d3d61ed5e1eb9","family_id":"qf-distance-squared","seed":202610064023,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"22\",\"23\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(10, -17) and Q=(32, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"1013","explanation":"Add the squares of the two coordinate differences to obtain 1013. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0024","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d822840f90f091e3696d5a0ebba41cd7bbed4b6711993359f42bef2ec40053fa","family_id":"qf-distance-squared","seed":202610064024,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"23\",\"24\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(11, -18) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"1105","explanation":"Add the squares of the two coordinate differences to obtain 1105. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0025","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ccf22617e98680f4e8b3a37efa10c1a888a3516b7eba57f8ef155bf45ef5ab7","family_id":"qf-distance-squared","seed":202610064025,"semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"24\"]],\"operation\":\"squared_distance\"}"},"fields":{"title":"Squared Euclidean Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 5). Find d(P,Q)² for the Euclidean L2 distance.","answer":{"kind":"short_text","value":"576","explanation":"Add the squares of the two coordinate differences to obtain 576. This is the squared distance, not its root; coincidence legitimately gives zero."},"skills":["Squared Euclidean Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0026","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7374d4f610ac2055bd41f3ed2f19b5f62c752cec649775068cbfc890ae97e3a9","family_id":"qf-distance-perfect-square","seed":202610064026,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"0\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-12, 5) and Q=(-12, 5). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"0","explanation":"The squared distance is 0, and its nonnegative principal root is 0. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0027","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36a5c33db95d96b932dfbc9ba5eae08bc3460d362599ab6cb73318fa977f4166","family_id":"qf-distance-perfect-square","seed":202610064027,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"6\",\"8\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-11, 4) and Q=(-19, 10). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"10","explanation":"The squared distance is 100, and its nonnegative principal root is 10. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0028","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65c353d66ba3c79260708d617e0672ab87063138d29565cc7ff374753f5e2124","family_id":"qf-distance-perfect-square","seed":202610064028,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"3\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-10, 3) and Q=(-10, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"3","explanation":"The squared distance is 9, and its nonnegative principal root is 3. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0029","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4b2b5df1886a3e16c6dccf21d7d0e2e1a664e7511df1f8e15003d7f4dd303f9","family_id":"qf-distance-perfect-square","seed":202610064029,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"10\",\"24\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-9, 2) and Q=(-33, 12). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"26","explanation":"The squared distance is 676, and its nonnegative principal root is 26. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0030","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fceb8fa8ae17290a248d78806120d71735f1f012b1d414a27a638c48c3de578","family_id":"qf-distance-perfect-square","seed":202610064030,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"5\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"5","explanation":"The squared distance is 25, and its nonnegative principal root is 5. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0031","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e65e1fde2977677ca71c8692a05cef3460a0d3ddf9aeba66e5c191f659eac557","family_id":"qf-distance-perfect-square","seed":202610064031,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"14\",\"48\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-55, 14). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"50","explanation":"The squared distance is 2500, and its nonnegative principal root is 50. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0032","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a841cad721cbc5cebf1579d104eb4daf5215a98860e99c2ce57b5c94700f849","family_id":"qf-distance-perfect-square","seed":202610064032,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"7\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(-6, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"7","explanation":"The squared distance is 49, and its nonnegative principal root is 7. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0033","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95b491270d3a858f1e54e418fe178a7760b2406d26e95456a7575b371ae9fd4d","family_id":"qf-distance-perfect-square","seed":202610064033,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"18\",\"80\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-5, -2) and Q=(-85, 16). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"82","explanation":"The squared distance is 6724, and its nonnegative principal root is 82. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0034","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc7a5a74725f599c2c277532035478267155d9fe1900fb73838afe066a91fc31","family_id":"qf-distance-perfect-square","seed":202610064034,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"9\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-4, -3) and Q=(-4, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"9","explanation":"The squared distance is 81, and its nonnegative principal root is 9. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0035","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5979c34b319aeb4b36394b358bde054dfe1599d597987911c84a3281d8fc2b95","family_id":"qf-distance-perfect-square","seed":202610064035,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"22\",\"120\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-123, 18). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"122","explanation":"The squared distance is 14884, and its nonnegative principal root is 122. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0036","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da7198cf3ff9be24aab3594580f73bd3042aa2f9a86cfb09eebb6952cb347e17","family_id":"qf-distance-perfect-square","seed":202610064036,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"11\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-2, -5) and Q=(-2, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"11","explanation":"The squared distance is 121, and its nonnegative principal root is 11. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0037","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcd78a4c30900b8fb5a585c25f108b5df2a97a269b076ceffd071e4e07611b1d","family_id":"qf-distance-perfect-square","seed":202610064037,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"26\",\"168\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-1, -6) and Q=(-169, 20). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"170","explanation":"The squared distance is 28900, and its nonnegative principal root is 170. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0038","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b76fd2ad65deb2b2a3ee52c03e0e6debee0f99657b28b8949d2279b8177e5afa","family_id":"qf-distance-perfect-square","seed":202610064038,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"13\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(0, -7) and Q=(0, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"13","explanation":"The squared distance is 169, and its nonnegative principal root is 13. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0039","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad2bf29dd4da61cf7afda63642e1ab43dbbb291dcae45dfb94a5d9ad1d332ade","family_id":"qf-distance-perfect-square","seed":202610064039,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"30\",\"224\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-223, 22). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"226","explanation":"The squared distance is 51076, and its nonnegative principal root is 226. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0040","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7477bd29245d8ea18371ff350dbd65365840e14fa784a52faa57db3b1ff6a88","family_id":"qf-distance-perfect-square","seed":202610064040,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"15\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(2, -9) and Q=(2, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"15","explanation":"The squared distance is 225, and its nonnegative principal root is 15. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0041","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0188d5337ea8562234b74166bc931f78bb9ff248aab9c8e4bc4dde5e5a829a8f","family_id":"qf-distance-perfect-square","seed":202610064041,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"34\",\"288\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-285, 24). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"290","explanation":"The squared distance is 84100, and its nonnegative principal root is 290. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0042","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"535fd3ad72d64410b12cfc33acf754bbbaee0dd17582191323460845cb136bb2","family_id":"qf-distance-perfect-square","seed":202610064042,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"17\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"17","explanation":"The squared distance is 289, and its nonnegative principal root is 17. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0043","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bb384f5291406fe1a70057b03c5aa7a5abe748018dca0ea60c593482e82ef71","family_id":"qf-distance-perfect-square","seed":202610064043,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"38\",\"360\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(5, -12) and Q=(-355, 26). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"362","explanation":"The squared distance is 131044, and its nonnegative principal root is 362. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0044","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c978cb966eaec063fe3d04595f928a3b6ba4831a60b6bd86984de3b8c2d7436","family_id":"qf-distance-perfect-square","seed":202610064044,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"19\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(6, -13) and Q=(6, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"19","explanation":"The squared distance is 361, and its nonnegative principal root is 19. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0045","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79eb22c0aaeec4a3ba7d6b86367296989b3871d1c1e63a866c4b9fec0feec3bc","family_id":"qf-distance-perfect-square","seed":202610064045,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"42\",\"440\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-433, 28). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"442","explanation":"The squared distance is 195364, and its nonnegative principal root is 442. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0046","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"addaed9984df88ce6c715e75d2cac9eba701e6ea5ed98cf565e37ae43cb1108d","family_id":"qf-distance-perfect-square","seed":202610064046,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"21\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(8, -15) and Q=(8, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"21","explanation":"The squared distance is 441, and its nonnegative principal root is 21. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0047","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a5fb73ed345accd582569e162820fa8626b22b628cb431f64138aeffcfd508c","family_id":"qf-distance-perfect-square","seed":202610064047,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"46\",\"528\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(9, -16) and Q=(-519, 30). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"530","explanation":"The squared distance is 280900, and its nonnegative principal root is 530. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0048","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9b4bf2c3792ad26b636690b2056b8be6fbf6ce085ade4a106854eb5d73e3a7f","family_id":"qf-distance-perfect-square","seed":202610064048,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"23\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(10, -17) and Q=(10, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"23","explanation":"The squared distance is 529, and its nonnegative principal root is 23. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0049","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bf2e21de8c08eccd72d94606416ffc00c8e4124819f4b6d01beb6585a25bc27","family_id":"qf-distance-perfect-square","seed":202610064049,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"50\",\"624\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(11, -18) and Q=(-613, 32). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"626","explanation":"The squared distance is 391876, and its nonnegative principal root is 626. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0050","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92ffdfbf6fc665b34cff9fc8f5488fef741c5627dfa5f121a823b51cea4aa080","family_id":"qf-distance-perfect-square","seed":202610064050,"semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"25\"]],\"operation\":\"perfect_square_distance\"}"},"fields":{"title":"Nonnegative Distance in a Perfect-Square Slice","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.","answer":{"kind":"short_text","value":"25","explanation":"The squared distance is 625, and its nonnegative principal root is 25. A negative root is not a distance, and no floating approximation is used."},"skills":["Nonnegative Distance in a Perfect-Square Slice"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0051","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67fc7f99f7a2dad7859278391683189c6f5af94c84c90d8d9f8889a8f3c6ed81","family_id":"qf-distance-candidate","seed":202610064051,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"0\"],\"0\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-12, 5) and Q=(-12, 5). Is the candidate d=0 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 0. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0052","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c3f39de906699e2563fb468c428041e559b14d05bad5cc7a172bc00abc2065b","family_id":"qf-distance-candidate","seed":202610064052,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"6\",\"8\"],\"-10\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-11, 4) and Q=(-19, 10). Is the candidate d=-10 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 100. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0053","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bd69e6a7f193ad5635863f2a27ead987ea7fcc6144a380494a72ab4a0e4de50","family_id":"qf-distance-candidate","seed":202610064053,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"3\"],\"4\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-10, 3) and Q=(-10, 6). Is the candidate d=4 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 9. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0054","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af690d2dc51358b40c1f0374434f51c778e31a2e67c08e8af5790209b5847515","family_id":"qf-distance-candidate","seed":202610064054,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"10\",\"24\"],\"26\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-9, 2) and Q=(-33, 12). Is the candidate d=26 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 676. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0055","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"271b190e63c48d783a5c2ee7216f8c021984fca93f4e3b01eeb5c0356a6b03bb","family_id":"qf-distance-candidate","seed":202610064055,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"5\"],\"5\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 6). Is the candidate d=5 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 25. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0056","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac7b5ad54536914663657581329c3d9ae1f27685457795697f14cf2b406ed3d","family_id":"qf-distance-candidate","seed":202610064056,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"14\",\"48\"],\"-50\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-55, 14). Is the candidate d=-50 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 2500. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0057","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"626df1f5652bb1439023a74cdc5d73fb1e7abc469115dc4af28d1f2e4c45b045","family_id":"qf-distance-candidate","seed":202610064057,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"7\"],\"8\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(-6, 6). Is the candidate d=8 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 49. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0058","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20572d665ac370841f521b7d6a2c5f76d816de9f0e1dec3b4a108f70ccbd40a7","family_id":"qf-distance-candidate","seed":202610064058,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"18\",\"80\"],\"82\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-5, -2) and Q=(-85, 16). Is the candidate d=82 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 6724. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0059","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8226ff282c1a2b1a9605311021f0c709ad41a9940644f2919c04dcba85c71cb7","family_id":"qf-distance-candidate","seed":202610064059,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"9\"],\"9\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-4, -3) and Q=(-4, 6). Is the candidate d=9 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 81. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0060","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"541360aa6a1e252f72613060c37096f92340e693f64fa64738763f133a6a47ce","family_id":"qf-distance-candidate","seed":202610064060,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"22\",\"120\"],\"-122\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-123, 18). Is the candidate d=-122 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 14884. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0061","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c37817c57d14bc4d2b593ca841fbf5cf138e614c63c3a6b8778f9e8d8242da0a","family_id":"qf-distance-candidate","seed":202610064061,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"11\"],\"12\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-2, -5) and Q=(-2, 6). Is the candidate d=12 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 121. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0062","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"992e246998efbcb5fe23a2e436f25a63463b062390d845b979d4084605e186c6","family_id":"qf-distance-candidate","seed":202610064062,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"26\",\"168\"],\"170\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-1, -6) and Q=(-169, 20). Is the candidate d=170 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 28900. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0063","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2f151885525e77307881b971e689086c30000a0e68feb4b4103ffa335040484","family_id":"qf-distance-candidate","seed":202610064063,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"13\"],\"13\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(0, -7) and Q=(0, 6). Is the candidate d=13 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 169. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0064","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"430d24fe4b562923a7377c67bea099854cc35307b56f3d21610151c155d4d5a9","family_id":"qf-distance-candidate","seed":202610064064,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"30\",\"224\"],\"-226\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-223, 22). Is the candidate d=-226 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 51076. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0065","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d27376b499f51cdc60b7b28b227214b89d54cd47b1ed06124d93f79f8c02b58","family_id":"qf-distance-candidate","seed":202610064065,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"15\"],\"16\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(2, -9) and Q=(2, 6). Is the candidate d=16 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 225. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0066","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3eaa69ca7b0bafe6692c82f1751fca96f3822c19e62d57fc18cbc8e39319561","family_id":"qf-distance-candidate","seed":202610064066,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"34\",\"288\"],\"290\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-285, 24). Is the candidate d=290 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 84100. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0067","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"752842c703afe76302ec8ffc1e04ce73b16d324c175b6537ad74572d5f800f58","family_id":"qf-distance-candidate","seed":202610064067,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"17\"],\"17\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 6). Is the candidate d=17 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 289. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0068","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75873ebbda9d5853ab9de2e5096b8faee13116b61fd5b25e89c6369607119510","family_id":"qf-distance-candidate","seed":202610064068,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"38\",\"360\"],\"-362\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(5, -12) and Q=(-355, 26). Is the candidate d=-362 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 131044. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0069","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c9d7f9a6a9af63355b15a61280b2fda27735da786cb720657d8b5f1964fbc85","family_id":"qf-distance-candidate","seed":202610064069,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"19\"],\"20\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(6, -13) and Q=(6, 6). Is the candidate d=20 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 361. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0070","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67d67912cb9a0a2cad3807e9c5d1f1deca1c9f6e27fc0c63b9b223a6e05adebd","family_id":"qf-distance-candidate","seed":202610064070,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"42\",\"440\"],\"442\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-433, 28). Is the candidate d=442 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 195364. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0071","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4f0143b2854be1e679c15b417548c73b526ef23e97c96eee08eb3c596fc26b7","family_id":"qf-distance-candidate","seed":202610064071,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"21\"],\"21\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(8, -15) and Q=(8, 6). Is the candidate d=21 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 441. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0072","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cee3362c90f09f34e390943e26cf1fd0356b1c40fba2a952ca146453d1f1285","family_id":"qf-distance-candidate","seed":202610064072,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"46\",\"528\"],\"-530\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(9, -16) and Q=(-519, 30). Is the candidate d=-530 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 280900. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0073","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b4ae247429f4bb5235267b06ceb0f0ad13fea3116d6e67c0bce7d6589eb131d","family_id":"qf-distance-candidate","seed":202610064073,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"23\"],\"24\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(10, -17) and Q=(10, 6). Is the candidate d=24 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"No","explanation":"The squared distance is 529. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0074","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2071a65c333f72fe4bd38f94ba4fe63bedc4495b621992500f069f9bc09b183d","family_id":"qf-distance-candidate","seed":202610064074,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"50\",\"624\"],\"626\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(11, -18) and Q=(-613, 32). Is the candidate d=626 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 391876. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0075","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfbc637e5c965d5fcb259b1139bc7a34897c6da79c9f813dbe515df40aa4c119","family_id":"qf-distance-candidate","seed":202610064075,"semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"25\"],\"25\"],\"operation\":\"candidate_distance\"}"},"fields":{"title":"Sign and Square Eligibility of a Candidate Distance","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 6). Is the candidate d=25 the Euclidean L2 distance? Check both its sign and its square.","answer":{"kind":"short_text","value":"Yes","explanation":"The squared distance is 625. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees."},"skills":["Sign and Square Eligibility of a Candidate Distance"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0076","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"154fa0f63792fbf3d6c64172fb2d8fef4eb43d2de08cf7eb6c8d5def45cd91d3","family_id":"qf-distance-comparison","seed":202610064076,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"0\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-12, 5) and Q=(-12, 5). In the same frame, U=(5, -3) and V=(5, -3). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 0 and 0 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0077","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"139cf3c669b96f7a035171032312a07d96fd6837d57b0184c0d7fa107b13a7ab","family_id":"qf-distance-comparison","seed":202610064077,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"1\",\"2\"],[\"1\",\"4\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-11, 4) and Q=(-12, 6). In the same frame, U=(4, -4) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"<","explanation":"Both distances are nonnegative, so compare their squares 5 and 17 exactly. The relation is <, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0078","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52d5535b5a13577b41530dee76ff80f3ce75cb91ee457ed8959cd9645016662a","family_id":"qf-distance-comparison","seed":202610064078,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"2\",\"3\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-10, 3) and Q=(-8, 6). In the same frame, U=(3, -5) and V=(3, -5). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":">","explanation":"Both distances are nonnegative, so compare their squares 13 and 0 exactly. The relation is >, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0079","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfb720abea5ab51d110269156692ab4907853cda03ea3b1a220eb64954e3fb96","family_id":"qf-distance-comparison","seed":202610064079,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"3\",\"4\"],[\"3\",\"4\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-9, 2) and Q=(-12, 6). In the same frame, U=(2, -6) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 25 and 25 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0080","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f64363994af0267a68a7b096e581103350b1c6b0dab27420f05f556cee4dc27","family_id":"qf-distance-comparison","seed":202610064080,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"4\"],[\"0\",\"6\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 5). In the same frame, U=(1, -7) and V=(1, -1). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"<","explanation":"Both distances are nonnegative, so compare their squares 16 and 36 exactly. The relation is <, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0081","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7752d401a5d1b65226dd3c3b4e52e99bc2b067c239fc4a296c0ba4051724192e","family_id":"qf-distance-comparison","seed":202610064081,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"5\",\"6\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-12, 6). In the same frame, U=(0, -8) and V=(0, -8). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":">","explanation":"Both distances are nonnegative, so compare their squares 61 and 0 exactly. The relation is >, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0082","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5fc99f94f9ae3fd1b4e0dc16ddcb97d9c6b94a700720963b08aa7d75c2161c9","family_id":"qf-distance-comparison","seed":202610064082,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"6\",\"7\"],[\"6\",\"7\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(0, 6). In the same frame, U=(-1, -9) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 85 and 85 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0083","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63b590788daa9fff00d7cd42c7ba98c9b56b2c53ae34b9fe33bbcb71b7dcace4","family_id":"qf-distance-comparison","seed":202610064083,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"7\",\"8\"],[\"7\",\"10\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-5, -2) and Q=(-12, 6). In the same frame, U=(-2, -10) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"<","explanation":"Both distances are nonnegative, so compare their squares 113 and 149 exactly. The relation is <, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0084","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a91469bfba5a23191b31f468d7f7e94ca66b41c674753b10584eacd57380e281","family_id":"qf-distance-comparison","seed":202610064084,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"8\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-4, -3) and Q=(-4, 5). In the same frame, U=(-3, -11) and V=(-3, -11). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":">","explanation":"Both distances are nonnegative, so compare their squares 64 and 0 exactly. The relation is >, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0085","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4db9611f2ad7abf06e91949dd352dd8e54e645e9406baaed0ccbc98c02026cf6","family_id":"qf-distance-comparison","seed":202610064085,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"9\",\"10\"],[\"9\",\"10\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-12, 6). In the same frame, U=(-4, -12) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 181 and 181 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0086","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7db4c19a98bded576e17a4dec7c07342df23464ad77cdf4c870eff7811c9d45b","family_id":"qf-distance-comparison","seed":202610064086,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"10\",\"11\"],[\"10\",\"13\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-2, -5) and Q=(8, 6). In the same frame, U=(-5, -13) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"<","explanation":"Both distances are nonnegative, so compare their squares 221 and 269 exactly. The relation is <, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0087","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d9210dd9f59f33533411496843bc0274cbcc827c9e50bdb3427cfd5f9b3bccb","family_id":"qf-distance-comparison","seed":202610064087,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"11\",\"12\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-1, -6) and Q=(-12, 6). In the same frame, U=(-6, -14) and V=(-6, -14). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":">","explanation":"Both distances are nonnegative, so compare their squares 265 and 0 exactly. The relation is >, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0088","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07076e65b29a981317b92d8cbb17005db448e1646f16e402d712ae8f9e83ec35","family_id":"qf-distance-comparison","seed":202610064088,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"12\"],[\"0\",\"12\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(0, -7) and Q=(0, 5). In the same frame, U=(-7, -15) and V=(-7, -3). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 144 and 144 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0089","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a53034fcb4b76eb40d9a5af4b0ae07df3315917b2e3fa2793150db4c590db7ba","family_id":"qf-distance-comparison","seed":202610064089,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"13\",\"14\"],[\"13\",\"16\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-12, 6). In the same frame, U=(-8, -16) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"<","explanation":"Both distances are nonnegative, so compare their squares 365 and 425 exactly. The relation is <, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0090","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78207ac667c8ff81d0ced101549d184a87ddfb322e49dc7d59f6a69720ce3fe6","family_id":"qf-distance-comparison","seed":202610064090,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"14\",\"15\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(2, -9) and Q=(16, 6). In the same frame, U=(-9, -17) and V=(-9, -17). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":">","explanation":"Both distances are nonnegative, so compare their squares 421 and 0 exactly. The relation is >, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0091","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df55b05c8efc5a0c009b2f09488d2f58433174fda0a4d969bef560bfac2789c6","family_id":"qf-distance-comparison","seed":202610064091,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"15\",\"16\"],[\"15\",\"16\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-12, 6). In the same frame, U=(-10, -18) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 481 and 481 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0092","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"110a22989a6bf8f44df8f7d054d702629d98c232af2148535b88859a446f24b2","family_id":"qf-distance-comparison","seed":202610064092,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"16\"],[\"0\",\"18\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 5). In the same frame, U=(-11, -19) and V=(-11, -1). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"<","explanation":"Both distances are nonnegative, so compare their squares 256 and 324 exactly. The relation is <, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0093","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41eb752b156eafc3f2414c4cc4172254b2d827356428f0adca7a699d5f191c56","family_id":"qf-distance-comparison","seed":202610064093,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"17\",\"18\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(5, -12) and Q=(-12, 6). In the same frame, U=(-12, -20) and V=(-12, -20). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":">","explanation":"Both distances are nonnegative, so compare their squares 613 and 0 exactly. The relation is >, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0094","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee4059353f6a81f155875869815ed9dae6014326e01bfabac866e6afa0abddc","family_id":"qf-distance-comparison","seed":202610064094,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"18\",\"19\"],[\"18\",\"19\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(6, -13) and Q=(24, 6). In the same frame, U=(-13, -21) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 685 and 685 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0095","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b87e072e84281109169e741e4cebd2a684deed0f5d1057e83f28d83fd8f8d0b","family_id":"qf-distance-comparison","seed":202610064095,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"19\",\"20\"],[\"19\",\"22\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-12, 6). In the same frame, U=(-14, -22) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"<","explanation":"Both distances are nonnegative, so compare their squares 761 and 845 exactly. The relation is <, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0096","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83d4680dd131b9afcc0498f7d7f2c8674096c39c59a88b692d0d9968d336a86f","family_id":"qf-distance-comparison","seed":202610064096,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"20\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(8, -15) and Q=(8, 5). In the same frame, U=(-15, -23) and V=(-15, -23). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":">","explanation":"Both distances are nonnegative, so compare their squares 400 and 0 exactly. The relation is >, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0097","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b009ed39b7871e2d38b31e9384c64335831a5c5dd13d4d060b53e0e07c45f8fe","family_id":"qf-distance-comparison","seed":202610064097,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"21\",\"22\"],[\"21\",\"22\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(9, -16) and Q=(-12, 6). In the same frame, U=(-16, -24) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 925 and 925 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0098","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7a23f86a3132d874c43d3a89fadd7e40521834195b039e4cbd05230f129a18e","family_id":"qf-distance-comparison","seed":202610064098,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"22\",\"23\"],[\"22\",\"25\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(10, -17) and Q=(32, 6). In the same frame, U=(-17, -25) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"<","explanation":"Both distances are nonnegative, so compare their squares 1013 and 1109 exactly. The relation is <, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0099","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"538351e59c4f56bf48a8730bfb8370cb6a8a2b618125aa920964315a225a06a0","family_id":"qf-distance-comparison","seed":202610064099,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"23\",\"24\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(11, -18) and Q=(-12, 6). In the same frame, U=(-18, -26) and V=(-18, -26). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":">","explanation":"Both distances are nonnegative, so compare their squares 1105 and 0 exactly. The relation is >, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261005-advance-20261006-r04-distance-formula-0100","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e6203476626ed9ddf9d27f93bea69a9d91292cb53295283eab5f023491c3310","family_id":"qf-distance-comparison","seed":202610064100,"semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"24\"],[\"0\",\"24\"]],\"operation\":\"distance_comparison\"}"},"fields":{"title":"Exact Comparison of Two Euclidean Distances","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 5). In the same frame, U=(-19, -27) and V=(-19, -3). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.","answer":{"kind":"short_text","value":"=","explanation":"Both distances are nonnegative, so compare their squares 576 and 576 exactly. The relation is =, without approximating square roots."},"skills":["Exact Comparison of Two Euclidean Distances"],"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."}}