{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r06-euclidean-length-0001","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63ad0f9f3a083ffe6a2ddcb6656edf652b5d268459c23f7a08db37569814aa3c","family_id":"qf-density12-length-displacement-square","seed":202610126001,"semantic_template":"qf-density12-length-displacement-square/[\"0\",\"0\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[0, 0]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(0, 0, 0)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (0, 0, 0)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0002","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbfd9e0149e004cb7169eeeeb9408ed137dcd5e469b6fb33ca770311f367ab88","family_id":"qf-density12-length-displacement-square","seed":202610126002,"semantic_template":"qf-density12-length-displacement-square/[\"-11\",\"-2\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 0], Q=[-12, -2]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-11, -2, 125)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-11, -2, 125)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0003","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"658979096141eba892b514b6496dedf6d58e97345a0eda1b220db5b74b02a042","family_id":"qf-density12-length-displacement-square","seed":202610126003,"semantic_template":"qf-density12-length-displacement-square/[\"-10\",\"3\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, -1], Q=[-10, 2]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-10, 3, 109)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-10, 3, 109)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0004","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a8a34a0545ca27bca47e3f43ca34df6ce5a24b49e99cbc1a1ce6113c027d887","family_id":"qf-density12-length-displacement-square","seed":202610126004,"semantic_template":"qf-density12-length-displacement-square/[\"-9\",\"-4\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 1], Q=[-8, -3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-9, -4, 97)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-9, -4, 97)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0005","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"844a757d1fcf6ee6ff07b0553099c625831cfa70c968a205681cef2074d09678","family_id":"qf-density12-length-displacement-square","seed":202610126005,"semantic_template":"qf-density12-length-displacement-square/[\"-8\",\"5\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 0], Q=[-10, 5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-8, 5, 89)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-8, 5, 89)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0006","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aec49539982bda27fbeb73add44ce8b825824368d7917470aff09900baf46f53","family_id":"qf-density12-length-displacement-square","seed":202610126006,"semantic_template":"qf-density12-length-displacement-square/[\"-7\",\"-1\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, -1], Q=[-8, -2]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-7, -1, 50)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-7, -1, 50)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0007","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9aea7313c19e37826f62bf3728015c685c43d5e17d125d6b8fc56b1273f7347f","family_id":"qf-density12-length-displacement-square","seed":202610126007,"semantic_template":"qf-density12-length-displacement-square/[\"-6\",\"2\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 1], Q=[-6, 3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-6, 2, 40)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-6, 2, 40)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0008","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"120754a1220ccd2c5a6ce44d8dcbb4a35f91ff9eb952c26a994feedb213e8fa3","family_id":"qf-density12-length-displacement-square","seed":202610126008,"semantic_template":"qf-density12-length-displacement-square/[\"-5\",\"-3\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 0], Q=[-4, -3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-5, -3, 34)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-5, -3, 34)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0009","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a790792ba7d4eb335545ea4546a2585eb1275f1cbd1465f5152c1660f6e3a45","family_id":"qf-density12-length-displacement-square","seed":202610126009,"semantic_template":"qf-density12-length-displacement-square/[\"-4\",\"4\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, -1], Q=[-6, 3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-4, 4, 32)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-4, 4, 32)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0010","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e96c866929748d77051296f1b42c04e826f186f6fbef17a92400383f5dc1bc63","family_id":"qf-density12-length-displacement-square","seed":202610126010,"semantic_template":"qf-density12-length-displacement-square/[\"-3\",\"-5\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 1], Q=[-4, -4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-3, -5, 34)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-3, -5, 34)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0011","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b51fd5db2e539f2fd7c186a3d3f14395c7c47b73abfb9128b9df8a2b77b1896e","family_id":"qf-density12-length-displacement-square","seed":202610126011,"semantic_template":"qf-density12-length-displacement-square/[\"-2\",\"1\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[-2, 1]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-2, 1, 5)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-2, 1, 5)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0012","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e8821e8c63af92b6f83e5b0af31db6c5746ec9c991ca1c541e29ae108b274a7","family_id":"qf-density12-length-displacement-square","seed":202610126012,"semantic_template":"qf-density12-length-displacement-square/[\"-1\",\"-2\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, -1], Q=[0, -3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(-1, -2, 5)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-1, -2, 5)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0013","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abe372bdd2cbc18f4a230c04954d89a2244dca4a411ec3e682c98e44f07a5df7","family_id":"qf-density12-length-displacement-square","seed":202610126013,"semantic_template":"qf-density12-length-displacement-square/[\"0\",\"3\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 1], Q=[-2, 4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(0, 3, 9)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (0, 3, 9)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0014","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f847f181d52848bac19c9f9b71ec48fb1337197775a90b9351253dea9f7a3eba","family_id":"qf-density12-length-displacement-square","seed":202610126014,"semantic_template":"qf-density12-length-displacement-square/[\"1\",\"-4\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 0], Q=[0, -4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(1, -4, 17)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (1, -4, 17)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0015","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43534fa676dfb9ecf25941cd44c8502f4684ed3f7ec37cced291ac6638122aa8","family_id":"qf-density12-length-displacement-square","seed":202610126015,"semantic_template":"qf-density12-length-displacement-square/[\"2\",\"5\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, -1], Q=[2, 4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(2, 5, 29)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (2, 5, 29)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0016","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f3c48d28967866b2cf37f1236e33991c15524525953441175f886408bae4532","family_id":"qf-density12-length-displacement-square","seed":202610126016,"semantic_template":"qf-density12-length-displacement-square/[\"3\",\"-1\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 1], Q=[4, 0]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(3, -1, 10)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (3, -1, 10)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0017","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"569a26a926855c9707020ac42c29d36e63c72a42f131bb0f5f71d299a4cdedc6","family_id":"qf-density12-length-displacement-square","seed":202610126017,"semantic_template":"qf-density12-length-displacement-square/[\"4\",\"2\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 0], Q=[2, 2]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(4, 2, 20)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (4, 2, 20)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0018","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cc501e0128cf3f0cc3d4d21a5c0a0237d1dfe7493c705fc23d0a361d3b61944","family_id":"qf-density12-length-displacement-square","seed":202610126018,"semantic_template":"qf-density12-length-displacement-square/[\"5\",\"-3\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, -1], Q=[4, -4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(5, -3, 34)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (5, -3, 34)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0019","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02de110050cad1d3ff1ac7f83956026641bbb52240d6b182088ee67d3df607c8","family_id":"qf-density12-length-displacement-square","seed":202610126019,"semantic_template":"qf-density12-length-displacement-square/[\"6\",\"4\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 1], Q=[6, 5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(6, 4, 52)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (6, 4, 52)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0020","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9540e126440aba85e685202a43e66323ba0111f6f71868c4bdbc3bb2aafd060","family_id":"qf-density12-length-displacement-square","seed":202610126020,"semantic_template":"qf-density12-length-displacement-square/[\"7\",\"-5\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 0], Q=[8, -5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(7, -5, 74)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (7, -5, 74)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0021","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06e3de0615dc96e44d12d39b74c0ad3b89d528ecb1c37686c984f70ae6e9435c","family_id":"qf-density12-length-displacement-square","seed":202610126021,"semantic_template":"qf-density12-length-displacement-square/[\"8\",\"1\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, -1], Q=[6, 0]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(8, 1, 65)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (8, 1, 65)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0022","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d432f476c3f6ccf96cc62a1328908fd95c226fdd84c67729de5abc61a97c5fe3","family_id":"qf-density12-length-displacement-square","seed":202610126022,"semantic_template":"qf-density12-length-displacement-square/[\"9\",\"-2\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 1], Q=[8, -1]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(9, -2, 85)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (9, -2, 85)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0023","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1bcc0dd0fb5a8b9f73f877ee88b073ec7a511a00afadbf3df30b333dc3333bd","family_id":"qf-density12-length-displacement-square","seed":202610126023,"semantic_template":"qf-density12-length-displacement-square/[\"10\",\"3\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[10, 3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(10, 3, 109)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (10, 3, 109)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0024","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ab14c0f0289b157d782074e4d9b4d3febc9875f91ee9d99657d3fc2876e4f86","family_id":"qf-density12-length-displacement-square","seed":202610126024,"semantic_template":"qf-density12-length-displacement-square/[\"11\",\"-4\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, -1], Q=[12, -5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(11, -4, 137)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (11, -4, 137)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0025","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fb27c74a49bf66f74eb8be74a959cf27b0843fa05ea7d448575924d8387a83f","family_id":"qf-density12-length-displacement-square","seed":202610126025,"semantic_template":"qf-density12-length-displacement-square/[\"12\",\"5\"]"},"fields":{"title":"Directed Components and Squared Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 1], Q=[10, 6]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.","answer":{"kind":"expression","value":"(12, 5, 169)","explanation":"Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (12, 5, 169)."},"skills":["Directed Components and Squared Length"],"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-20261012-r06-euclidean-length-0026","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ad7ede74a4912cbc8647cf4a13fdc344064bda42fdbc4ae674f36a449b72b8b","family_id":"qf-density12-length-integer-distance-pair","seed":202610126026,"semantic_template":"qf-density12-length-integer-distance-pair/[\"0\",\"0\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [0, 0], [0, 0]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (0, 0)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0027","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43e35fbaf174917c6434b7a313e2675785e815e615c6e2521f283f7021cb72b6","family_id":"qf-density12-length-integer-distance-pair","seed":202610126027,"semantic_template":"qf-density12-length-integer-distance-pair/[\"25\",\"0\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [3, 4], [3, 4]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(5, 0)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (5, 0)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0028","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6781e155ef01046110a66dc52e43db4e879365444f42b2dda17080c65d4481a1","family_id":"qf-density12-length-integer-distance-pair","seed":202610126028,"semantic_template":"qf-density12-length-integer-distance-pair/[\"0\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [0, 0], [5, 12]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(0, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (0, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0029","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b704e4d30362279bdbf90c9a5cc3f2fb0033c6fe6cc3415c6403f1ce5a463be0","family_id":"qf-density12-length-integer-distance-pair","seed":202610126029,"semantic_template":"qf-density12-length-integer-distance-pair/[\"25\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [3, 4], [8, 16]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(5, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (5, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0030","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4222c597e445e69c6b3c173d53c6a3d5ea32b8e02a086751e4854c54f154ad89","family_id":"qf-density12-length-integer-distance-pair","seed":202610126030,"semantic_template":"qf-density12-length-integer-distance-pair/[\"100\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [6, 8], [11, 20]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(10, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (10, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0031","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92147a42d753d8b5fb7fdae09f08072e3bc7c2a3af7d1120c7f77113c3661a77","family_id":"qf-density12-length-integer-distance-pair","seed":202610126031,"semantic_template":"qf-density12-length-integer-distance-pair/[\"225\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [9, 12], [14, 24]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(15, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (15, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0032","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18d21e851bd9cee053fc8f676158d0a17d69acd73cb2c19f5ac22a14788c69c5","family_id":"qf-density12-length-integer-distance-pair","seed":202610126032,"semantic_template":"qf-density12-length-integer-distance-pair/[\"400\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [12, 16], [17, 28]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(20, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (20, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0033","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb06ceeac0ecc0dc1524514fd64712399eb220fcbba7088a868c194b668924f6","family_id":"qf-density12-length-integer-distance-pair","seed":202610126033,"semantic_template":"qf-density12-length-integer-distance-pair/[\"625\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [15, 20], [20, 32]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(25, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (25, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0034","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ea41fedb421e9a929af3c11f53423fafe63dc191367a9cac940e03333af3643","family_id":"qf-density12-length-integer-distance-pair","seed":202610126034,"semantic_template":"qf-density12-length-integer-distance-pair/[\"900\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [18, 24], [23, 36]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(30, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (30, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0035","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b15fe3605f5d9ef1393d5c049d298e7bdc84a59e9dcbbbee49fc5f70e99e1b","family_id":"qf-density12-length-integer-distance-pair","seed":202610126035,"semantic_template":"qf-density12-length-integer-distance-pair/[\"1225\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [21, 28], [26, 40]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(35, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (35, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0036","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d399cb33f8b3951d1ca92e41707c05ddf56698d02b8269c06b27e11c296308c","family_id":"qf-density12-length-integer-distance-pair","seed":202610126036,"semantic_template":"qf-density12-length-integer-distance-pair/[\"1600\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [24, 32], [29, 44]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(40, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (40, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0037","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7ba7fbe2344b9e0d2e76b4a8c03cddb89437c97a2ebd9d2d3dcbd0721c125d4","family_id":"qf-density12-length-integer-distance-pair","seed":202610126037,"semantic_template":"qf-density12-length-integer-distance-pair/[\"2025\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [27, 36], [32, 48]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(45, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (45, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0038","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b48124b962561ff4cf76af068086619a6070b9f3dca8217dddca68a30077fcd","family_id":"qf-density12-length-integer-distance-pair","seed":202610126038,"semantic_template":"qf-density12-length-integer-distance-pair/[\"2500\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [30, 40], [35, 52]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(50, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (50, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0039","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e2d8ae112a08ba8fa5e13ff06a614a847a223a841d7bb5315a208d5051f077c","family_id":"qf-density12-length-integer-distance-pair","seed":202610126039,"semantic_template":"qf-density12-length-integer-distance-pair/[\"3025\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [33, 44], [38, 56]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(55, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (55, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0040","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ee6f66562fe80f040b8dbcac88be8781bd819cc570d65fd47e0bcaab29e550a","family_id":"qf-density12-length-integer-distance-pair","seed":202610126040,"semantic_template":"qf-density12-length-integer-distance-pair/[\"3600\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [36, 48], [41, 60]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(60, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (60, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0041","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca3e2aea34a252fc2afec8cf006077fd6bd79b6c55722ea64f04a442028b99b7","family_id":"qf-density12-length-integer-distance-pair","seed":202610126041,"semantic_template":"qf-density12-length-integer-distance-pair/[\"4225\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [39, 52], [44, 64]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(65, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (65, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0042","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66eefe5fb32c6062e107946b4f4c20f5ff6efce5a3435edb71d9754e54cb255d","family_id":"qf-density12-length-integer-distance-pair","seed":202610126042,"semantic_template":"qf-density12-length-integer-distance-pair/[\"4900\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [42, 56], [47, 68]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(70, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (70, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0043","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7d9b0da5a3667491a3cde1a9d30cceaa1b6fd1870006e4c028a48dd16b0b092","family_id":"qf-density12-length-integer-distance-pair","seed":202610126043,"semantic_template":"qf-density12-length-integer-distance-pair/[\"5625\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [45, 60], [50, 72]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(75, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (75, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0044","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b8a1691b080dd28b2da6733605c671dbd8700e7c71c4aad15d66775d51f8935","family_id":"qf-density12-length-integer-distance-pair","seed":202610126044,"semantic_template":"qf-density12-length-integer-distance-pair/[\"6400\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [48, 64], [53, 76]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(80, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (80, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0045","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"785d92daa566d88f04a99432929ce7904abd9fab880fd9d03ed58025b6239f79","family_id":"qf-density12-length-integer-distance-pair","seed":202610126045,"semantic_template":"qf-density12-length-integer-distance-pair/[\"7225\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [51, 68], [56, 80]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(85, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (85, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0046","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ca48b38b2641197699f8f6e85d3f56fe1a145dfa3734540fa090cf8ca140009","family_id":"qf-density12-length-integer-distance-pair","seed":202610126046,"semantic_template":"qf-density12-length-integer-distance-pair/[\"8100\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [54, 72], [59, 84]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(90, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (90, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0047","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84f6dc1bde71950d4779ba93f89f9a959047f2362367b04a3f100ff8fc4ea54","family_id":"qf-density12-length-integer-distance-pair","seed":202610126047,"semantic_template":"qf-density12-length-integer-distance-pair/[\"9025\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [57, 76], [62, 88]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(95, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (95, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0048","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"360f04fac34e23a589c42af066e9687d6f28cf6d86c31b7d3b7af2174ffcf93c","family_id":"qf-density12-length-integer-distance-pair","seed":202610126048,"semantic_template":"qf-density12-length-integer-distance-pair/[\"10000\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [60, 80], [65, 92]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(100, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (100, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0049","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c256b8cd1f24e2340bbc4f8555ef833d2e78880a7bf94a2e85c207bd708fc9e","family_id":"qf-density12-length-integer-distance-pair","seed":202610126049,"semantic_template":"qf-density12-length-integer-distance-pair/[\"11025\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [63, 84], [68, 96]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(105, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (105, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0050","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd68d73186f1484192d1f8b2cc60ca54c97b1f50fa9621615e93b6c3d6d699f2","family_id":"qf-density12-length-integer-distance-pair","seed":202610126050,"semantic_template":"qf-density12-length-integer-distance-pair/[\"12100\",\"169\"]"},"fields":{"title":"Ordered Integer Principal Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [66, 88], [71, 100]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.","answer":{"kind":"expression","value":"(110, 13)","explanation":"Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (110, 13)."},"skills":["Ordered Integer Principal Segment Lengths"],"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-20261012-r06-euclidean-length-0051","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9b953c89199db6abea7107e3ee579631f56abc8fa19db89c83904f8ceba9f65","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126051,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"0/1\",\"0/1\",\"0\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[0, 0], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 0/1, 0/1, 0/1)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (0/1, 0/1, 0/1, 0/1)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0052","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4423312a3d600fbcd505a8ea74938d93f965216671c36bdb271ba992ffca6467","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126052,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"0/1\",\"1\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[1, 0], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 1/4, 1/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 0/1, 1/4, 1/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0053","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3923b57734a5b74cda4f9cf965706b3c1e3702fc35663e029aae0f78e0e93966","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126053,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"5/2\",\"34\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 1], Q=[3, 4], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 5/2, 17/2, 17/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 5/2, 17/2, 17/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0054","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dbce53ffaa807b2abb7e8e99750a13b67f1839340894ee1f3fc8f600984ac7a","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126054,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"3/1\",\"65\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-3, 1], Q=[4, 5], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 3/1, 65/4, 65/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 3/1, 65/4, 65/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0055","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9909eecadd5d8f1734a4314d463d63e808d77d2524382a7dd59ea9f18a1dac1","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126055,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"7/2\",\"106\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-4, 1], Q=[5, 6], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 7/2, 53/2, 53/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 7/2, 53/2, 53/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0056","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"212e0d9e6dd13954fd69b00b0372d3ae7ac824e700e2c6099f82d7916c66c516","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126056,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"4/1\",\"157\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-5, 1], Q=[6, 7], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 4/1, 157/4, 157/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 4/1, 157/4, 157/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0057","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7f63466df65e05124e8b16bfe4dd959e2632283b7120c0490e224f4a971320a","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126057,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"9/2\",\"218\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-6, 1], Q=[7, 8], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 9/2, 109/2, 109/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 9/2, 109/2, 109/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0058","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59bfa85a2f302bef429ecc686cb3a9f6739fe827dc9eb7692f7dd134768cb142","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126058,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"5/1\",\"289\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-7, 1], Q=[8, 9], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 5/1, 289/4, 289/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 5/1, 289/4, 289/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0059","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff5ffadb6bd4d59f0531999521cbc43b7736977b78f1a31f03a0b91582c909e1","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126059,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"11/2\",\"370\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-8, 1], Q=[9, 10], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 11/2, 185/2, 185/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 11/2, 185/2, 185/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0060","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8182aef26f1aec9a85fd6234d7f401f62588daafee3f706103a06b426a9fcf7","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126060,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"6/1\",\"461\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-9, 1], Q=[10, 11], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 6/1, 461/4, 461/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 6/1, 461/4, 461/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0061","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d9f487f0ec9fed4bcf4912b6f9d087c9072e10aa304f0a7b94cb63b83df7e15","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126061,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"13/2\",\"562\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-10, 1], Q=[11, 12], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 13/2, 281/2, 281/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 13/2, 281/2, 281/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0062","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a86a0189e60fcdbe97eb4c0e6a384deec59b1bc41fce58179a4212af4c510d32","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126062,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"7/1\",\"673\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-11, 1], Q=[12, 13], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 7/1, 673/4, 673/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 7/1, 673/4, 673/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0063","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e77fd36122308248effbe816c5276002f7bb8d3fd314e77a8391a8d4e856af5e","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126063,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"15/2\",\"794\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-12, 1], Q=[13, 14], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 15/2, 397/2, 397/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 15/2, 397/2, 397/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0064","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0fcb6d0bdf677d96c7ec90a807bb01e40d6c9b412d36f3d6a9d6062382b55d4","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126064,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"8/1\",\"925\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-13, 1], Q=[14, 15], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 8/1, 925/4, 925/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 8/1, 925/4, 925/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0065","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30a53879a1f2cc0e19a76783d217155de95b488cc9d1aada3f7d335f8d0cdac5","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126065,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"17/2\",\"1066\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-14, 1], Q=[15, 16], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 17/2, 533/2, 533/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 17/2, 533/2, 533/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0066","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5cbb3af20264e1a6371577699615715613e298f5725d500b83cd402aa2ff518","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126066,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"9/1\",\"1217\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-15, 1], Q=[16, 17], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 9/1, 1217/4, 1217/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 9/1, 1217/4, 1217/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0067","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d3fb92ab69e4b49b0d8008a29caf509ca6eda10f116302f17997cfb6455749a","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126067,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"19/2\",\"1378\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-16, 1], Q=[17, 18], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 19/2, 689/2, 689/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 19/2, 689/2, 689/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0068","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa3b3bc832f20c5d416ce268cf8df06a22ef62e32a06fd128863902c8547a03c","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126068,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"10/1\",\"1549\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-17, 1], Q=[18, 19], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 10/1, 1549/4, 1549/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 10/1, 1549/4, 1549/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0069","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e31e2c41a926ab348ada3e6fb5ed0ba2320f55ec5119b788034df94a84fa7484","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126069,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"21/2\",\"1730\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-18, 1], Q=[19, 20], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 21/2, 865/2, 865/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 21/2, 865/2, 865/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0070","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0f939f7eca01e7788fb0fb563b21097b3a147522fe06e4e079c52d5d593b80c","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126070,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"11/1\",\"1921\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-19, 1], Q=[20, 21], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 11/1, 1921/4, 1921/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 11/1, 1921/4, 1921/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0071","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9dbfd1a6de5c53bc2982f685787860b938e5550e4e257b678e5d244d1df326b","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126071,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"23/2\",\"2122\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-20, 1], Q=[21, 22], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 23/2, 1061/2, 1061/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 23/2, 1061/2, 1061/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0072","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06b0ee9def1fad018983b6305e7c5f6a8d168d1814af99a196d613599095d018","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126072,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"12/1\",\"2333\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-21, 1], Q=[22, 23], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 12/1, 2333/4, 2333/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 12/1, 2333/4, 2333/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0073","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7772f4fe577065e5c5cade7e0ed002880a5fd2d887b54735fdc11dd3271f1d52","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126073,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"25/2\",\"2554\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-22, 1], Q=[23, 24], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 25/2, 1277/2, 1277/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 25/2, 1277/2, 1277/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0074","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"169383e91042e10b2e75184be5b9bd95c78add83a128a6ad26b17f7741ae7af3","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126074,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"13/1\",\"2785\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-23, 1], Q=[24, 25], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 13/1, 2785/4, 2785/4)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 13/1, 2785/4, 2785/4)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0075","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39faec46f50a54ed7735006fac47cc08a008da92ba9383d590c23673e3ad67b4","family_id":"qf-density12-length-midpoint-equidistance","seed":202610126075,"semantic_template":"qf-density12-length-midpoint-equidistance/[\"1/2\",\"27/2\",\"3026\"]"},"fields":{"title":"Exact Midpoint and Two Squared Segment Lengths","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-24, 1], Q=[25, 26], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 27/2, 1513/2, 1513/2)","explanation":"Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 27/2, 1513/2, 1513/2)."},"skills":["Exact Midpoint and Two Squared Segment Lengths"],"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-20261012-r06-euclidean-length-0076","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f9684f96251620c0bb51e3621744a0fdb5069ecce0ffd0287b703944d78844f","family_id":"qf-density12-length-triangle-defect","seed":202610126076,"semantic_template":"qf-density12-length-triangle-defect/[\"0\",\"0\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(0,0), R=(0,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(0, 0, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (0, 0, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0077","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ece2c3440d8a2431efba682a7f17cc4bdbd6dc566f5250fe3f9cfa46b0f64ae7","family_id":"qf-density12-length-triangle-defect","seed":202610126077,"semantic_template":"qf-density12-length-triangle-defect/[\"2\",\"2\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(1,0), R=(2,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(2, 2, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (2, 2, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0078","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f530a0a9f2818632cc3eab858c5ab8da1f810c92db78e0a9df3e6fafbae6c3b9","family_id":"qf-density12-length-triangle-defect","seed":202610126078,"semantic_template":"qf-density12-length-triangle-defect/[\"4\",\"2\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-1,0), R=(2,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(4, 2, 2, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (4, 2, 2, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0079","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51e85851070afd5dc03f42af58d61d6ed01de27848bfa898b92e30919afa17c6","family_id":"qf-density12-length-triangle-defect","seed":202610126079,"semantic_template":"qf-density12-length-triangle-defect/[\"9\",\"3\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-3,0), R=(3,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(9, 3, 6, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (9, 3, 6, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0080","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2329feae105813ac17dd716aa4e5e1ff2e1a6ec299c20bd7449756affb90877","family_id":"qf-density12-length-triangle-defect","seed":202610126080,"semantic_template":"qf-density12-length-triangle-defect/[\"4\",\"4\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(2,0), R=(4,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(4, 4, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (4, 4, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0081","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b5eea832d31e516b4f3d973b96da62ca8ed84197dcad8c08d09b8bb22a4562f","family_id":"qf-density12-length-triangle-defect","seed":202610126081,"semantic_template":"qf-density12-length-triangle-defect/[\"15\",\"5\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-5,0), R=(5,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(15, 5, 10, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (15, 5, 10, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0082","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7bc294226eed4fdcf841a83f255abed987183b3d8009fbc9497915a76b9a7a7","family_id":"qf-density12-length-triangle-defect","seed":202610126082,"semantic_template":"qf-density12-length-triangle-defect/[\"6\",\"6\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(3,0), R=(6,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(6, 6, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (6, 6, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0083","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"467113f7cc18b18cbafc38b7bfe4ad1522d8e3fd1fe3a63fe5e0e47115b53b11","family_id":"qf-density12-length-triangle-defect","seed":202610126083,"semantic_template":"qf-density12-length-triangle-defect/[\"21\",\"7\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-7,0), R=(7,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(21, 7, 14, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (21, 7, 14, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0084","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d9d3a48f9e5f41fd737656736604cb755650a5c27d9bc1479faf4cb945437f6","family_id":"qf-density12-length-triangle-defect","seed":202610126084,"semantic_template":"qf-density12-length-triangle-defect/[\"8\",\"8\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(4,0), R=(8,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(8, 8, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (8, 8, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0085","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0421aa7b3c775fb757554e3461f4937f656e27c2578a2effd4a39ec35de396a9","family_id":"qf-density12-length-triangle-defect","seed":202610126085,"semantic_template":"qf-density12-length-triangle-defect/[\"27\",\"9\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-9,0), R=(9,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(27, 9, 18, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (27, 9, 18, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0086","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99546fcdc25b26b712dc6c2d4c425079fcf50a0a87e4fe0ca1e96c4d0979e0d3","family_id":"qf-density12-length-triangle-defect","seed":202610126086,"semantic_template":"qf-density12-length-triangle-defect/[\"10\",\"10\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(5,0), R=(10,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(10, 10, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (10, 10, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0087","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fecae4b0080375246e3bc20da1e6b346e467b9a36362fe873d6423a23d491ae7","family_id":"qf-density12-length-triangle-defect","seed":202610126087,"semantic_template":"qf-density12-length-triangle-defect/[\"33\",\"11\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-11,0), R=(11,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(33, 11, 22, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (33, 11, 22, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0088","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e8ea793c038154381e6d7de7229848ed716f030489c52d63c19baa6145f24f","family_id":"qf-density12-length-triangle-defect","seed":202610126088,"semantic_template":"qf-density12-length-triangle-defect/[\"12\",\"12\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(6,0), R=(12,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(12, 12, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (12, 12, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0089","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b15b1b7050ed2d58726106433162fa7d5ea7a79b8e3f95b2c6d3cd3c0c87664e","family_id":"qf-density12-length-triangle-defect","seed":202610126089,"semantic_template":"qf-density12-length-triangle-defect/[\"39\",\"13\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-13,0), R=(13,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(39, 13, 26, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (39, 13, 26, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0090","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d48b8eb99b979a6d7c765749715d2d6c55667f4311e08907333c10a6da0b3e9","family_id":"qf-density12-length-triangle-defect","seed":202610126090,"semantic_template":"qf-density12-length-triangle-defect/[\"14\",\"14\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(7,0), R=(14,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(14, 14, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (14, 14, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0091","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a861f966341d923ae37a1b86a649bde3625d027ae013688e6440de01ddc86d3c","family_id":"qf-density12-length-triangle-defect","seed":202610126091,"semantic_template":"qf-density12-length-triangle-defect/[\"45\",\"15\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-15,0), R=(15,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(45, 15, 30, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (45, 15, 30, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0092","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61b744fbc2c09c57890af16037716daf2863f50f01c191769b67c35727a09756","family_id":"qf-density12-length-triangle-defect","seed":202610126092,"semantic_template":"qf-density12-length-triangle-defect/[\"16\",\"16\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(8,0), R=(16,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(16, 16, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (16, 16, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0093","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c4c3d2d21fb504e9f5ef3c41e9033b54fee6958a7d166a1329733556cee847e","family_id":"qf-density12-length-triangle-defect","seed":202610126093,"semantic_template":"qf-density12-length-triangle-defect/[\"51\",\"17\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-17,0), R=(17,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(51, 17, 34, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (51, 17, 34, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0094","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5d9c79d161dbd9dc19e499a56ff527380ff603b81ba1e71cea17e13a184a58f","family_id":"qf-density12-length-triangle-defect","seed":202610126094,"semantic_template":"qf-density12-length-triangle-defect/[\"18\",\"18\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(9,0), R=(18,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(18, 18, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (18, 18, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0095","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d38ebf2b08d0d4c5d2bfd9218fa268b16bb657e40fc56f710d7488dd12a42475","family_id":"qf-density12-length-triangle-defect","seed":202610126095,"semantic_template":"qf-density12-length-triangle-defect/[\"57\",\"19\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-19,0), R=(19,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(57, 19, 38, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (57, 19, 38, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0096","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d79248e53cd633d1ab54d151262d34de50b22fbae8193159b22152242e122d0b","family_id":"qf-density12-length-triangle-defect","seed":202610126096,"semantic_template":"qf-density12-length-triangle-defect/[\"20\",\"20\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(10,0), R=(20,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(20, 20, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (20, 20, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0097","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2050ec5da4f07e86921b3ffcc2a01cc603f99aab10121971aa4dc92a4557d46","family_id":"qf-density12-length-triangle-defect","seed":202610126097,"semantic_template":"qf-density12-length-triangle-defect/[\"63\",\"21\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-21,0), R=(21,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(63, 21, 42, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (63, 21, 42, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0098","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce9c692a1e016f8c23542d8608cc04b3b506bd6652fe223a8fa2e302bf3cedee","family_id":"qf-density12-length-triangle-defect","seed":202610126098,"semantic_template":"qf-density12-length-triangle-defect/[\"22\",\"22\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(11,0), R=(22,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(22, 22, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (22, 22, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0099","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc081f06833d1a7cff63170c8c31231f400a5f3e091f81925e532acf55b3f21f","family_id":"qf-density12-length-triangle-defect","seed":202610126099,"semantic_template":"qf-density12-length-triangle-defect/[\"69\",\"23\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-23,0), R=(23,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(69, 23, 46, No)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (69, 23, 46, No)."},"skills":["Axis Path Sum versus Endpoint Length"],"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-20261012-r06-euclidean-length-0100","batch_id":"batch-20261008-advance-20261012-r06-euclidean-length-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"688dfa6119e3ab256c06ad086b603a0f9ac669c2963d6b4592b728fdaf7794ad","family_id":"qf-density12-length-triangle-defect","seed":202610126100,"semantic_template":"qf-density12-length-triangle-defect/[\"24\",\"24\"]"},"fields":{"title":"Axis Path Sum versus Endpoint Length","stem":"Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(12,0), R=(24,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).","answer":{"kind":"expression","value":"(24, 24, 0, Yes)","explanation":"Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (24, 24, 0, Yes)."},"skills":["Axis Path Sum versus Endpoint Length"],"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."}}