{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r03-distance-formula-0001","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60b2903e1dfd84490c4d1b633940bd653b8105b1b3d5e79b9484c2d1c7530958","family_id":"qf-density10-distance-squares-pair","seed":202610103001,"semantic_template":"qf-density10-distance-squares-pair/[[\"0\",\"0\"],[\"3\",\"4\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [3, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(0, 25)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (0, 25)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0002","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b92dbc794c76bef7243dc5edf42d39ef4974ecc6ee6ce46964eb170976478b2","family_id":"qf-density10-distance-squares-pair","seed":202610103002,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"1\"],[\"2\",\"3\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[1, -1], [2, 3]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(2, 13)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (2, 13)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0003","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97fd771f7afc8e6fe0940b867c73522511803eddc6e4719da23ffbdd6f3c9e86","family_id":"qf-density10-distance-squares-pair","seed":202610103003,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"2\"],[\"3\",\"4\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[2, -1], [3, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(5, 25)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (5, 25)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0004","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b8da25de89e17bec7eb01163b6566551892d077b2055f7bb10a5d5a0d05999b","family_id":"qf-density10-distance-squares-pair","seed":202610103004,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"3\"],[\"2\",\"4\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[3, -1], [4, 2]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(10, 20)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (10, 20)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0005","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df4e12571853b16cc9fa3c82c76d0b7e0796ac79610585c2b78d2ea78cf2949a","family_id":"qf-density10-distance-squares-pair","seed":202610103005,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"4\"],[\"3\",\"5\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[4, -1], [5, 3]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(17, 34)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (17, 34)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0006","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e54e261941a2ab92f04b0ec15c864158116f0afd9d900c3f802272d02977d323","family_id":"qf-density10-distance-squares-pair","seed":202610103006,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"5\"],[\"4\",\"6\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[5, -1], [6, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(26, 52)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (26, 52)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0007","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6259b09fa3a1a9891d5e02ca0f96752e747f3f633d3a3eb9effaa53371621366","family_id":"qf-density10-distance-squares-pair","seed":202610103007,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"6\"],[\"2\",\"7\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[6, -1], [7, 2]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(37, 53)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (37, 53)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0008","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb6b1c03878dd1c60ab4f4d676313303b1fc92ad0e235132abfa0c5c6f356677","family_id":"qf-density10-distance-squares-pair","seed":202610103008,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"7\"],[\"3\",\"8\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[7, -1], [8, 3]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(50, 73)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (50, 73)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0009","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d3a038a9b1e00a57c7f8509d9214eb858c7c338bc85dd6c849943a43fa8ed0a","family_id":"qf-density10-distance-squares-pair","seed":202610103009,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"8\"],[\"4\",\"9\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[8, -1], [9, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(65, 97)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (65, 97)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0010","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc5927705e225a70b2996d2b8e1e6eb4b00e5851782bbe7806aa54d74829fb73","family_id":"qf-density10-distance-squares-pair","seed":202610103010,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"9\"],[\"10\",\"2\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[9, -1], [10, 2]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(82, 104)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (82, 104)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0011","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4253b3bbd6243e04189dbc515657f5699849a1b9052e20ab9833d9f491336fab","family_id":"qf-density10-distance-squares-pair","seed":202610103011,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"10\"],[\"11\",\"3\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[10, -1], [11, 3]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(101, 130)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (101, 130)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0012","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ae0809ac63efd8a234c6dc1f2c39fb854001b79c87593b12b9586152b791589","family_id":"qf-density10-distance-squares-pair","seed":202610103012,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"11\"],[\"12\",\"4\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[11, -1], [12, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(122, 160)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (122, 160)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0013","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa70c228e0b56657bc40fdea75f49c4f2c2968aacc0dd7402191fe3ecbef310e","family_id":"qf-density10-distance-squares-pair","seed":202610103013,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"12\"],[\"13\",\"2\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[12, -1], [13, 2]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(145, 173)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (145, 173)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0014","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e48d81f7e78f4299ca413ba422557fd26f618f142ba91ee719efaab22ed77e1e","family_id":"qf-density10-distance-squares-pair","seed":202610103014,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"13\"],[\"14\",\"3\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[13, -1], [14, 3]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(170, 205)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (170, 205)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0015","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a047608670377cba72ae5656f1fd5ddceafca1cb15e2d2327af9a69e50806cfc","family_id":"qf-density10-distance-squares-pair","seed":202610103015,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"14\"],[\"15\",\"4\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[14, -1], [15, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(197, 241)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (197, 241)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0016","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab30a53765552fb60cf2c7a133ae45c78243f3f3a6ee5222693cab3366e10dd1","family_id":"qf-density10-distance-squares-pair","seed":202610103016,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"15\"],[\"16\",\"2\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[15, -1], [16, 2]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(226, 260)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (226, 260)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0017","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e0c3fea06b8390229e333b333facb8a24ce36ef412073b272782c93d52c65f0","family_id":"qf-density10-distance-squares-pair","seed":202610103017,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"16\"],[\"17\",\"3\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[16, -1], [17, 3]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(257, 298)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (257, 298)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0018","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aec1d174556e47c936db16e41d6bc4d1e31cdda7afb2b2c69f50aae9f9089deb","family_id":"qf-density10-distance-squares-pair","seed":202610103018,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"17\"],[\"18\",\"4\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[17, -1], [18, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(290, 340)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (290, 340)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0019","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f1be57c82bc5dd57a18d72227f6bb647448586d22f0b0f64f1f910285df7c67","family_id":"qf-density10-distance-squares-pair","seed":202610103019,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"18\"],[\"19\",\"2\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[18, -1], [19, 2]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(325, 365)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (325, 365)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0020","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f63913df2b1625f7451bee0f19021ed10bce872255c4710079897e24772d2d59","family_id":"qf-density10-distance-squares-pair","seed":202610103020,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"19\"],[\"20\",\"3\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[19, -1], [20, 3]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(362, 409)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (362, 409)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0021","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a41be70c772e76e7e0f20ae1514b46eae21a013ff91691f83977f516cf8ecf10","family_id":"qf-density10-distance-squares-pair","seed":202610103021,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"20\"],[\"21\",\"4\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[20, -1], [21, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(401, 457)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (401, 457)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0022","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0639a124d1906bd7ce9fa2c28885823f38f7001fbc5b2084e50ecb9c30d9a76b","family_id":"qf-density10-distance-squares-pair","seed":202610103022,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"21\"],[\"2\",\"22\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[21, -1], [22, 2]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(442, 488)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (442, 488)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0023","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"246a01106a9dae1b85877044f2ac15d9435b3e6627c99e01c9bf169d1885c0cf","family_id":"qf-density10-distance-squares-pair","seed":202610103023,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"22\"],[\"23\",\"3\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[22, -1], [23, 3]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(485, 538)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (485, 538)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0024","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afafe99a174d735c6a5c4e63c3ca2c5505957c0e6046127a11ed9b19bc03b6a2","family_id":"qf-density10-distance-squares-pair","seed":202610103024,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"23\"],[\"24\",\"4\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[23, -1], [24, 4]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(530, 592)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (530, 592)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0025","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1284c18a4868455eab1215e78f29d89753e62c7704e98b1b7cf3dc6e6d5aac30","family_id":"qf-density10-distance-squares-pair","seed":202610103025,"semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"24\"],[\"2\",\"25\"]]"},"fields":{"title":"Ordered Pair of Squared Euclidean Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[24, -1], [25, 2]] in order. Give their two squared L2 distances.","answer":{"kind":"expression","value":"(577, 629)","explanation":"Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (577, 629)."},"skills":["Ordered Pair of Squared 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-20261008-advance-20261010-r03-distance-formula-0026","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c63cb8d06964d667282deb9b3d9fd32d70e03d65c5dabc7989a9a9bad8713af2","family_id":"qf-density10-distance-candidate-defect","seed":202610103026,"semantic_template":"qf-density10-distance-candidate-defect/[[\"3\",\"4\"],-5]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=-5. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, 0)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, 0)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0027","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ef842f77a8dfce0dc58051d07e56cf2ff73b96628aa357e3d6e13a2d6ce1023","family_id":"qf-density10-distance-candidate-defect","seed":202610103027,"semantic_template":"qf-density10-distance-candidate-defect/[[\"3\",\"4\"],5]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=5. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, 0)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 0)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0028","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6fe94c575a0a3313acbf3106d6cf4dfb813336eeb9c970e004bcaee0f7b5952","family_id":"qf-density10-distance-candidate-defect","seed":202610103028,"semantic_template":"qf-density10-distance-candidate-defect/[[\"3\",\"4\"],4]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=4. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -9)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -9)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0029","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9eed0119889dba9e807f354af0876bf054ad128a84c6cc426ca0ca2dcc1886d","family_id":"qf-density10-distance-candidate-defect","seed":202610103029,"semantic_template":"qf-density10-distance-candidate-defect/[[\"0\",\"0\"],0]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[0, 0], integer candidate c=0. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, 0)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 0)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0030","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"483633cdb3b25e9a02b1f459c7deb262de19890ac5aa317b4c3e2e994c64b5c9","family_id":"qf-density10-distance-candidate-defect","seed":202610103030,"semantic_template":"qf-density10-distance-candidate-defect/[[\"4\",\"5\"],7]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[4, 5], integer candidate c=7. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, 8)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 8)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0031","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7014c004dcc34bf705478f1264d92ff558eacb79874003ebf66424dc59219e9b","family_id":"qf-density10-distance-candidate-defect","seed":202610103031,"semantic_template":"qf-density10-distance-candidate-defect/[[\"5\",\"6\"],-8]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[5, 6], integer candidate c=-8. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, 3)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, 3)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0032","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"071bab56e2620630f15717b93b62fc5c21a2dbd2a44256e1157a423740df9528","family_id":"qf-density10-distance-candidate-defect","seed":202610103032,"semantic_template":"qf-density10-distance-candidate-defect/[[\"6\",\"7\"],9]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[6, 7], integer candidate c=9. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -4)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -4)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0033","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a14401698f2c4052c546f0d19e62efea43666351d011441f485843b390b2ae91","family_id":"qf-density10-distance-candidate-defect","seed":202610103033,"semantic_template":"qf-density10-distance-candidate-defect/[[\"7\",\"8\"],-10]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[7, 8], integer candidate c=-10. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -13)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -13)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0034","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ace0d68c48ce8aab36e8cd25124bfa90a947d3fe731cce7871adcde65309e8e","family_id":"qf-density10-distance-candidate-defect","seed":202610103034,"semantic_template":"qf-density10-distance-candidate-defect/[[\"8\",\"9\"],11]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[8, 9], integer candidate c=11. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -24)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -24)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0035","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beed7385f662e282854e995439ccaa7239eed25cd1957ea9c99f08ac6569cfaf","family_id":"qf-density10-distance-candidate-defect","seed":202610103035,"semantic_template":"qf-density10-distance-candidate-defect/[[\"10\",\"9\"],-12]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[9, 10], integer candidate c=-12. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -37)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -37)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0036","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d261fc42eeb94dff4d8dead85facf46f9092d7dce76a19463773807d360a2311","family_id":"qf-density10-distance-candidate-defect","seed":202610103036,"semantic_template":"qf-density10-distance-candidate-defect/[[\"10\",\"11\"],13]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[10, 11], integer candidate c=13. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -52)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -52)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0037","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"081c67097e2ddc9996cf9af670d71a58a1e6c4bdc223d8d71d454ab2b3584997","family_id":"qf-density10-distance-candidate-defect","seed":202610103037,"semantic_template":"qf-density10-distance-candidate-defect/[[\"11\",\"12\"],-14]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[11, 12], integer candidate c=-14. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -69)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -69)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0038","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc82324041f3138b979c3b5c3d70ddf69f4e5c71f6697cdcd5cb127b514eb41d","family_id":"qf-density10-distance-candidate-defect","seed":202610103038,"semantic_template":"qf-density10-distance-candidate-defect/[[\"12\",\"13\"],15]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[12, 13], integer candidate c=15. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -88)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -88)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0039","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e04cca86850036f859a24f231dc583703884ab8efb840f8ee2357faa4243b5e","family_id":"qf-density10-distance-candidate-defect","seed":202610103039,"semantic_template":"qf-density10-distance-candidate-defect/[[\"13\",\"14\"],-16]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[13, 14], integer candidate c=-16. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -109)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -109)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0040","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d170467859bfb95fe6031df18e7801b63ec404a5451be815c5d5036b8f2ed747","family_id":"qf-density10-distance-candidate-defect","seed":202610103040,"semantic_template":"qf-density10-distance-candidate-defect/[[\"14\",\"15\"],17]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[14, 15], integer candidate c=17. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -132)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -132)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0041","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3aec8ec7dcb779ec1542c08b7a4da3d4847f3e0628ff0647036d5782c5e10c3","family_id":"qf-density10-distance-candidate-defect","seed":202610103041,"semantic_template":"qf-density10-distance-candidate-defect/[[\"15\",\"16\"],-18]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[15, 16], integer candidate c=-18. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -157)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -157)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0042","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"321a90fa9d2d8d5050689f560cdbffc1e36083826d6a2a30549b254b1d17e1f6","family_id":"qf-density10-distance-candidate-defect","seed":202610103042,"semantic_template":"qf-density10-distance-candidate-defect/[[\"16\",\"17\"],19]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[16, 17], integer candidate c=19. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -184)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -184)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0043","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61614c737c8b4dc5e97095a6d8f3f40292fae88211df6b7c87a07ca2265fe12d","family_id":"qf-density10-distance-candidate-defect","seed":202610103043,"semantic_template":"qf-density10-distance-candidate-defect/[[\"17\",\"18\"],-20]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[17, 18], integer candidate c=-20. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -213)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -213)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0044","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d42e93d60c5ca2a984d03c9df63405e41f3ce81b188bcc58020856489012ed2d","family_id":"qf-density10-distance-candidate-defect","seed":202610103044,"semantic_template":"qf-density10-distance-candidate-defect/[[\"18\",\"19\"],21]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[18, 19], integer candidate c=21. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -244)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -244)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0045","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3a25e444d89883a0ba47e607bf12d45a610ebc3923968b7b812eac3f0db3ee9","family_id":"qf-density10-distance-candidate-defect","seed":202610103045,"semantic_template":"qf-density10-distance-candidate-defect/[[\"19\",\"20\"],-22]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[19, 20], integer candidate c=-22. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -277)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -277)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0046","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e8cf5c2ecf6817f709db593b3602a5b532f43eb7ed581edfb1256612c7a2613","family_id":"qf-density10-distance-candidate-defect","seed":202610103046,"semantic_template":"qf-density10-distance-candidate-defect/[[\"20\",\"21\"],23]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[20, 21], integer candidate c=23. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -312)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -312)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0047","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08c1a514cd0824b1a7f0d5603d31c63ba111f201ff0e98b2d9ed99590dcbfe3b","family_id":"qf-density10-distance-candidate-defect","seed":202610103047,"semantic_template":"qf-density10-distance-candidate-defect/[[\"21\",\"22\"],-24]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[21, 22], integer candidate c=-24. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -349)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -349)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0048","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdfec64b9819c2382650ea9b9a812b641ab9ddfc077f21dc9a3ed0ca4f722601","family_id":"qf-density10-distance-candidate-defect","seed":202610103048,"semantic_template":"qf-density10-distance-candidate-defect/[[\"22\",\"23\"],25]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[22, 23], integer candidate c=25. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -388)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -388)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0049","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"baa58ec21af0a4326deea3ec67fec274c33cb97b430f87735848e1dc2c8b3234","family_id":"qf-density10-distance-candidate-defect","seed":202610103049,"semantic_template":"qf-density10-distance-candidate-defect/[[\"23\",\"24\"],-26]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[23, 24], integer candidate c=-26. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(No, -429)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -429)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0050","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45c1081a246110368cf8c9776345790d2920de4f49536db90033ffd1384d23aa","family_id":"qf-density10-distance-candidate-defect","seed":202610103050,"semantic_template":"qf-density10-distance-candidate-defect/[[\"24\",\"25\"],27]"},"fields":{"title":"Candidate Sign and Squared-Distance Defect","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[24, 25], integer candidate c=27. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.","answer":{"kind":"expression","value":"(Yes, -472)","explanation":"Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -472)."},"skills":["Candidate Sign and Squared-Distance Defect"],"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-20261008-advance-20261010-r03-distance-formula-0051","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df0d9bd35e83950fe0b5e84310c0705370c24791a4980ef240b605442086f31b","family_id":"qf-density10-distance-three-order","seed":202610103051,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"0\"],[\"0\",\"0\"],[\"0\",\"0\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[0, 0], [0, 0], [0, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[1, 2, 3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1, 2, 3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0052","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd24fb01281bf2022bf2dffcb4cb39066e386be7283e3a633cb37fefcd90e537","family_id":"qf-density10-distance-three-order","seed":202610103052,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"1\"],[\"0\",\"1\"],[\"0\",\"2\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[1, 0], [0, -1], [2, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[1, 2], [3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1, 2], [3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0053","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"233b0f88c1b30f1361149e64b7952a40b40311ad07b9d1283667804fe2b1aed0","family_id":"qf-density10-distance-three-order","seed":202610103053,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"2\"],[\"0\",\"1\"],[\"1\",\"1\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[2, 0], [1, 0], [1, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2], [3], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2], [3], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0054","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0160e905debec5c6c5187a9e2c2ef4186414845cd4af246cf5e3b795c2230741","family_id":"qf-density10-distance-three-order","seed":202610103054,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"1\"],[\"0\",\"1\"],[\"1\",\"1\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[1, 1], [1, 0], [-1, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2], [1, 3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2], [1, 3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0055","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2cb99678a7dc19f7defe96c551902b41db718a0d372c791a3c9a1af67590108","family_id":"qf-density10-distance-three-order","seed":202610103055,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"2\"],[\"1\",\"2\"],[\"0\",\"0\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[2, 1], [1, 2], [0, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[3], [1, 2]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [1, 2]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0056","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"501d2f466cb52404e4b6611a7eac600f36705376aaa89e8d76903bd52cd73591","family_id":"qf-density10-distance-three-order","seed":202610103056,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"6\"],[\"1\",\"5\"],[\"0\",\"5\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[6, 0], [5, 1], [5, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0057","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c83993ee59033fa9eba375bbf3772a8ca7e23148176b643190440bc71d9dc354","family_id":"qf-density10-distance-three-order","seed":202610103057,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"7\"],[\"0\",\"6\"],[\"0\",\"6\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[7, 1], [6, 0], [0, -6]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0058","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f47ef5ac69c2e68fcbc7b6d76c2ebbedf8dd6c88d6cbcdb7d3d43237a852a93c","family_id":"qf-density10-distance-three-order","seed":202610103058,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"7\"],[\"0\",\"8\"],[\"1\",\"9\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[7, 0], [8, 0], [9, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0059","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3221e1dd82de4adaefdb598e3751309615e447fbb26fe5f3b8546891da22e222","family_id":"qf-density10-distance-three-order","seed":202610103059,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"9\"],[\"1\",\"8\"],[\"0\",\"8\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[9, 0], [8, 1], [8, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0060","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9182dad5318d138c32ed76cace07a5db3c7c6e2503cb7ff30c71b650ce323683","family_id":"qf-density10-distance-three-order","seed":202610103060,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"10\"],[\"0\",\"9\"],[\"0\",\"9\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[10, 1], [9, 0], [0, -9]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0061","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"250c9610bfacb1705a1fea05fbb065a777250fc4a292aea16529c78d8b8f54bf","family_id":"qf-density10-distance-three-order","seed":202610103061,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"10\"],[\"0\",\"11\"],[\"1\",\"12\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[10, 0], [11, 0], [12, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0062","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72d7ab85cd7a4ac72cc3396b080d6d12397c32322b89fe84a72cf01c30a38a92","family_id":"qf-density10-distance-three-order","seed":202610103062,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"12\"],[\"1\",\"11\"],[\"0\",\"11\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[12, 0], [11, 1], [11, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0063","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e776178824ccfcf2e56098036fe7df7f00d4034bb66a1cb363a1d0a1181808","family_id":"qf-density10-distance-three-order","seed":202610103063,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"13\"],[\"0\",\"12\"],[\"0\",\"12\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[13, 1], [12, 0], [0, -12]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0064","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5f5499e16c27429fa628386b93acd4fed41ba249cabe6a6d9824fd907f69669","family_id":"qf-density10-distance-three-order","seed":202610103064,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"13\"],[\"0\",\"14\"],[\"1\",\"15\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[13, 0], [14, 0], [15, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0065","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5fecc021a70bd9a133ba00db4b44a648320fa4e861d7cd279c4290d6a337d10","family_id":"qf-density10-distance-three-order","seed":202610103065,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"15\"],[\"1\",\"14\"],[\"0\",\"14\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[15, 0], [14, 1], [14, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0066","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb283f6f6284a7296e5e1c372d8bd29f01c32b156cb0021c74ece8e3cc2b6d45","family_id":"qf-density10-distance-three-order","seed":202610103066,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"16\"],[\"0\",\"15\"],[\"0\",\"15\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[16, 1], [15, 0], [0, -15]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0067","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4979bd60980e22789a31a77a077c3c7bc58f3917948503b1eece11fea43d2a64","family_id":"qf-density10-distance-three-order","seed":202610103067,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"16\"],[\"0\",\"17\"],[\"1\",\"18\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[16, 0], [17, 0], [18, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0068","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01b13b291e4713bffc4174a66c1379ccc1ef555fbd78ae472d1721ea72ae13a1","family_id":"qf-density10-distance-three-order","seed":202610103068,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"18\"],[\"1\",\"17\"],[\"0\",\"17\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[18, 0], [17, 1], [17, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0069","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b520664e16b3a398e27f0c8ad8dbdc868eaaff1c093150713145eb2c060f35a3","family_id":"qf-density10-distance-three-order","seed":202610103069,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"19\"],[\"0\",\"18\"],[\"0\",\"18\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[19, 1], [18, 0], [0, -18]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0070","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8efdbb9163a78e853d8e8a72c77cd72e997da8e5835c1e0f49f7a7dcf1100adf","family_id":"qf-density10-distance-three-order","seed":202610103070,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"19\"],[\"0\",\"20\"],[\"1\",\"21\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[19, 0], [20, 0], [21, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0071","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"274114eeb8a133ff3138b9f6dcb5d6fe60ef98de2aa621883c7265cdefb1cc2b","family_id":"qf-density10-distance-three-order","seed":202610103071,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"21\"],[\"1\",\"20\"],[\"0\",\"20\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[21, 0], [20, 1], [20, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0072","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffc09bdd8dcceca28d7017c20e5ffe232aaf609422a9c094d086532ac48600f4","family_id":"qf-density10-distance-three-order","seed":202610103072,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"22\"],[\"0\",\"21\"],[\"0\",\"21\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[22, 1], [21, 0], [0, -21]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0073","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"070a13fbac4ea0cb9974f43a81d8141d380c67d707be38b7225e011efd1fc5b2","family_id":"qf-density10-distance-three-order","seed":202610103073,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"22\"],[\"0\",\"23\"],[\"1\",\"24\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[22, 0], [23, 0], [24, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0074","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"353676dedcd03d99118d4e6625c02c14153597d33bcf5c00ea2e71571862bccf","family_id":"qf-density10-distance-three-order","seed":202610103074,"semantic_template":"qf-density10-distance-three-order/[[\"0\",\"24\"],[\"1\",\"23\"],[\"0\",\"23\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[24, 0], [23, 1], [23, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0075","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4caa0c27f158ea33aece0785eddf5fee4dbb8517e5f32ea8a970b9dee59053ff","family_id":"qf-density10-distance-three-order","seed":202610103075,"semantic_template":"qf-density10-distance-three-order/[[\"1\",\"25\"],[\"0\",\"24\"],[\"0\",\"24\"]]"},"fields":{"title":"Distance Order with Explicit Tie Groups","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[25, 1], [24, 0], [0, -24]]. List index groups in increasing distance order, with indices increasing inside each tie group.","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation":"Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]]."},"skills":["Distance Order with Explicit Tie Groups"],"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-20261008-advance-20261010-r03-distance-formula-0076","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a6f01e633b37a57a9741a640d155ba01f6b7634cd4d34363c23bcf3f5085675","family_id":"qf-density10-distance-integer-sum","seed":202610103076,"semantic_template":"qf-density10-distance-integer-sum/[[\"0\",\"0\"],[\"0\",\"0\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [0, 0]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"0","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 0."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0077","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"763186827833c6f558d6527812eca3d505da800559f7113a3d044a004d20b347","family_id":"qf-density10-distance-integer-sum","seed":202610103077,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"3\",\"4\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[3, 4], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"18","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 18."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0078","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7924de4d1ae00703e82c380976ad237bbe85a871a1a9e5697e455868b770e694","family_id":"qf-density10-distance-integer-sum","seed":202610103078,"semantic_template":"qf-density10-distance-integer-sum/[[\"0\",\"0\"],[\"3\",\"4\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [3, 4]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"5","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 5."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0079","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"921916270d8c94cd8263bcac192c78c11b938aa0caaa003f56bcdf586fe898f6","family_id":"qf-density10-distance-integer-sum","seed":202610103079,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"12\",\"9\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[9, 12], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"28","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 28."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0080","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4214ef5177fd4fac06b95a4c3a9342c49838954898a1047ccc3cc2b55d8dc69","family_id":"qf-density10-distance-integer-sum","seed":202610103080,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"16\"],[\"12\",\"5\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[12, 16], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"33","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 33."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0081","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d43a24f8e4d5e219c386479514131410959a9f8496faee5b14a1082be5aca80","family_id":"qf-density10-distance-integer-sum","seed":202610103081,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"15\",\"20\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[15, 20], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"38","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 38."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0082","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4059ed8769da0b4d94f9ef5b7334c6e0461116db4c79491a99f246a018612d2c","family_id":"qf-density10-distance-integer-sum","seed":202610103082,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"18\",\"24\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[18, 24], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"43","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 43."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0083","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39bdc6b668be1d16e280c11dd422899d8d819b210eca855d0348a97eae5b82fe","family_id":"qf-density10-distance-integer-sum","seed":202610103083,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"21\",\"28\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[21, 28], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"48","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 48."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0084","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71debc6847e182f9a2b524a31ee6f63045884f7768b7738c2f28f364b28fee8c","family_id":"qf-density10-distance-integer-sum","seed":202610103084,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"24\",\"32\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[24, 32], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"53","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 53."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0085","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"780aaaad15d6f738a19ba72b309a52768577d92c7a6465a5e2a8fce0e25746aa","family_id":"qf-density10-distance-integer-sum","seed":202610103085,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"27\",\"36\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[27, 36], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"58","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 58."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0086","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c2ece5ab202c6c6aa9ef931f93dc465c4b244d5b58343c58d19807119538c0f","family_id":"qf-density10-distance-integer-sum","seed":202610103086,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"30\",\"40\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[30, 40], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"63","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 63."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0087","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f582571308f3112eb0504b2afb46e32a1f54ecc4b9add16baac78bc0eecede22","family_id":"qf-density10-distance-integer-sum","seed":202610103087,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"33\",\"44\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[33, 44], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"68","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 68."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0088","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c37074692acbb25424cb6f2157635f53dd3d58efdec05ea2d70a7ecbaf2365ca","family_id":"qf-density10-distance-integer-sum","seed":202610103088,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"36\",\"48\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[36, 48], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"73","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 73."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0089","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d3987935c9be569ac0e2d8a9281e52deee25a4caa6545569d99343cc9cdabb0","family_id":"qf-density10-distance-integer-sum","seed":202610103089,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"39\",\"52\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[39, 52], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"78","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 78."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0090","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf4176f9abb62d7cfd4d6a64252290c925bb6c66441b6d6e753b0a07e1eb3b31","family_id":"qf-density10-distance-integer-sum","seed":202610103090,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"42\",\"56\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[42, 56], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"83","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 83."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0091","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7db34a4fa8abab3509e692773f41c56402fd3ef02daa09b1d0bb3ca49c88d5d9","family_id":"qf-density10-distance-integer-sum","seed":202610103091,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"45\",\"60\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[45, 60], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"88","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 88."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0092","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d5f154d80c25f28792a2dcd6931984d8396cd994d6cf9e6d40c5840958847f1","family_id":"qf-density10-distance-integer-sum","seed":202610103092,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"48\",\"64\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[48, 64], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"93","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 93."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0093","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8c92cd9d13bacbc8c2d810aecf584ca09edfba1740a33756b79c914a49cb2fe","family_id":"qf-density10-distance-integer-sum","seed":202610103093,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"51\",\"68\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[51, 68], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"98","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 98."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0094","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e75649e26dd4e4fe84757113dc8064d78587cb40504c4a670a683729ec547f8","family_id":"qf-density10-distance-integer-sum","seed":202610103094,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"54\",\"72\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[54, 72], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"103","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 103."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0095","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7ff67737e3c8edb562b5f02e8681d7fe254a8ee991a248692fd0c2c5234d673","family_id":"qf-density10-distance-integer-sum","seed":202610103095,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"57\",\"76\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[57, 76], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"108","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 108."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0096","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90d3b2bd383fa8fc3b6a56962dada4bcd0cd8036d984a3aab2f5d3ddce015b27","family_id":"qf-density10-distance-integer-sum","seed":202610103096,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"60\",\"80\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[60, 80], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"113","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 113."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0097","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dcd1eff4747981c1dfd29357877f820a9de14afd3a1fce66e706ffa4af2b1c0","family_id":"qf-density10-distance-integer-sum","seed":202610103097,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"63\",\"84\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[63, 84], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"118","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 118."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0098","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd48011ff8fb622f3091c03de6e1f82dec2fb6263d14264c9696d1b57d685985","family_id":"qf-density10-distance-integer-sum","seed":202610103098,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"66\",\"88\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[66, 88], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"123","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 123."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0099","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8afcac261afbb669f717af5135ecdf0d53bf763568f8eb41d7fe1b80db383e71","family_id":"qf-density10-distance-integer-sum","seed":202610103099,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"69\",\"92\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[69, 92], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"128","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 128."},"skills":["Sum of Two Integer Principal 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-20261008-advance-20261010-r03-distance-formula-0100","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b53080c28ac329ff49e781f2b8f82ebf9c8343811b7bb7115fb1799c3ab976a3","family_id":"qf-density10-distance-integer-sum","seed":202610103100,"semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"72\",\"96\"]]"},"fields":{"title":"Sum of Two Integer Principal Distances","stem":"In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[72, 96], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.","answer":{"kind":"expression","value":"133","explanation":"Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 133."},"skills":["Sum of Two Integer Principal 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."}}