{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0001","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7e99a9796fb7e730efcec1c13700255f7153df96e23017022c23e69d7fb2231","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147001,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"0\",[\"0\",\"1\",\"1\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 0], r=0, ordered points=[[0, 0], [1, 0], [0, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 1, 1], [0, 1, 1], [on-boundary, outside, outside], [Yes, No, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 1, 1], [0, 1, 1], [on-boundary, outside, outside], [Yes, No, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0002","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb85bf6d5f1cf2abff6cb8535fb46fe62cf5b0340c46cc75e776e62ca9e057d0","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147002,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"1\",[\"0\",\"1\",\"5\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 0], r=1, ordered points=[[0, 0], [1, 0], [2, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 1, 5], [-1, 0, 4], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 1, 5], [-1, 0, 4], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0003","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a34c88ce5d69265e75a070277fc078d81ceeb35cacd5edd6e44bc6593bfbc38e","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147003,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"4\",[\"0\",\"4\",\"10\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -1], r=2, ordered points=[[1, -1], [3, -1], [4, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 4, 10], [-4, 0, 6], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 4, 10], [-4, 0, 6], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0004","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f375c5c2f4e1e78e3d0ba07522d38e7cd10b4c2574104e41e70dfb55e92d6624","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147004,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"9\",[\"0\",\"9\",\"17\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, -2], r=3, ordered points=[[-1, -2], [2, -2], [3, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 9, 17], [-9, 0, 8], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 9, 17], [-9, 0, 8], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0005","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c8ed3020397fe9e5b276c42137edfebda70c40790295c5a9bf332dd3698c915","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147005,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"16\",[\"0\",\"16\",\"26\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 1], r=4, ordered points=[[0, 1], [4, 1], [5, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 16, 26], [-16, 0, 10], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 16, 26], [-16, 0, 10], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0006","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89142d1e0ad8df21fac7b9f9215599bb31f30477909517c71ff0a96ba1409e9c","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147006,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"25\",[\"0\",\"25\",\"37\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, 0], r=5, ordered points=[[1, 0], [6, 0], [7, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 25, 37], [-25, 0, 12], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 25, 37], [-25, 0, 12], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0007","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0ed2c5601f020f387b82d3e73c41c9660960cf57d3f36a77fc2a857c3981ad1","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147007,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"36\",[\"0\",\"36\",\"50\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, -1], r=6, ordered points=[[-1, -1], [5, -1], [6, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 36, 50], [-36, 0, 14], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 36, 50], [-36, 0, 14], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0008","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73eaeb8ed70c6c3b9cf3e627b8bb55c51be6d5849526dacccd5873fa4de2afa5","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147008,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"49\",[\"0\",\"49\",\"65\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, -2], r=7, ordered points=[[0, -2], [7, -2], [8, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 49, 65], [-49, 0, 16], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 49, 65], [-49, 0, 16], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0009","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"455413fb1327e4481b8962895cbb69ac9b11f8a1d5769ae20cc8f8006407e14f","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147009,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"64\",[\"0\",\"64\",\"82\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, 1], r=8, ordered points=[[1, 1], [9, 1], [10, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 64, 82], [-64, 0, 18], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 64, 82], [-64, 0, 18], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0010","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62a5bfe3fe77ab4e42e6ff6ab4bdeb101bc5e0b442b0c278a0d910c707f032e8","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147010,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"81\",[\"0\",\"81\",\"101\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, 0], r=9, ordered points=[[-1, 0], [8, 0], [9, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 81, 101], [-81, 0, 20], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 81, 101], [-81, 0, 20], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0011","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aa54b63022cf5a8dfb1a14fd306d9eb29c5e169b3160a365fe3aee90600be15","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147011,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"100\",[\"0\",\"100\",\"122\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, -1], r=10, ordered points=[[0, -1], [10, -1], [11, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 100, 122], [-100, 0, 22], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 100, 122], [-100, 0, 22], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0012","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74067e1b5f2192cde77109846c860bfb60f916c6dc910e1faf8c2f11d89fbd9a","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147012,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"121\",[\"0\",\"121\",\"145\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -2], r=11, ordered points=[[1, -2], [12, -2], [13, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 121, 145], [-121, 0, 24], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 121, 145], [-121, 0, 24], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0013","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93c4cad31bd7c724c46ab2cfe3ac250d55d0e3a605fa2cfefa1bda4257f088cb","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147013,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"144\",[\"0\",\"144\",\"170\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, 1], r=12, ordered points=[[-1, 1], [11, 1], [12, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 144, 170], [-144, 0, 26], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 144, 170], [-144, 0, 26], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0014","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56639e6ded020430f42a7a470168ab275a454fb260a78e257bdda0c3be7c44ca","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147014,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"169\",[\"0\",\"169\",\"197\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 0], r=13, ordered points=[[0, 0], [13, 0], [14, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 169, 197], [-169, 0, 28], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 169, 197], [-169, 0, 28], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0015","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53562df1e0d384ec3af21585c8e6fe355eed19788e5a25ffea4aa0fa4b182e69","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147015,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"196\",[\"0\",\"196\",\"226\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -1], r=14, ordered points=[[1, -1], [15, -1], [16, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 196, 226], [-196, 0, 30], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 196, 226], [-196, 0, 30], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0016","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1a66439514cb5f09d6d46d2dff87659eed3dbe2e1ba1819b253c9ec4b0c4779","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147016,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"225\",[\"0\",\"225\",\"257\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, -2], r=15, ordered points=[[-1, -2], [14, -2], [15, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 225, 257], [-225, 0, 32], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 225, 257], [-225, 0, 32], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0017","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb4102732697470e395c27e3f4bb9fab0e687d591e51dbd5ec9df6574af06403","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147017,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"256\",[\"0\",\"256\",\"290\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 1], r=16, ordered points=[[0, 1], [16, 1], [17, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 256, 290], [-256, 0, 34], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 256, 290], [-256, 0, 34], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0018","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2de3af4d4c699c0288bbd32c61dd4861c395f78e64bd35c39b84f61acf1c3b10","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147018,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"289\",[\"0\",\"289\",\"325\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, 0], r=17, ordered points=[[1, 0], [18, 0], [19, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 289, 325], [-289, 0, 36], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 289, 325], [-289, 0, 36], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0019","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"545464738488785797fc3ce34a7330260e878cd504149c6d7ca49045d21298e2","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147019,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"324\",[\"0\",\"324\",\"362\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, -1], r=18, ordered points=[[-1, -1], [17, -1], [18, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 324, 362], [-324, 0, 38], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 324, 362], [-324, 0, 38], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0020","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8aad32dc696d0b58644cfdf2138b085d37473004c74eec52bd483f51fb514974","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147020,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"361\",[\"0\",\"361\",\"401\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, -2], r=19, ordered points=[[0, -2], [19, -2], [20, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 361, 401], [-361, 0, 40], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 361, 401], [-361, 0, 40], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0021","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c6b1d1201ff4b898dc66d49bf6eb1490479679a4cde0e4dc0a7878a6e461c18","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147021,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"400\",[\"0\",\"400\",\"442\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, 1], r=20, ordered points=[[1, 1], [21, 1], [22, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 400, 442], [-400, 0, 42], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 400, 442], [-400, 0, 42], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0022","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7021bcb693c44baba4f8ab120f296c2864b1b07fce3f34bf92fe21ba208f4f3b","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147022,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"441\",[\"0\",\"441\",\"485\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, 0], r=21, ordered points=[[-1, 0], [20, 0], [21, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 441, 485], [-441, 0, 44], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 441, 485], [-441, 0, 44], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0023","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3173cdddf35d40eaa20e6b76048a339ddd12ca5c68889742999b209ba291655e","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147023,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"484\",[\"0\",\"484\",\"530\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, -1], r=22, ordered points=[[0, -1], [22, -1], [23, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 484, 530], [-484, 0, 46], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 484, 530], [-484, 0, 46], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0024","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67b01b78a78d4d20ec2e8f87cb697cff002584eb1205110962e39812cadd33e5","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147024,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"529\",[\"0\",\"529\",\"577\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -2], r=23, ordered points=[[1, -2], [24, -2], [25, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 529, 577], [-529, 0, 48], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 529, 577], [-529, 0, 48], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0025","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9733bd59e540beafbdeb144e21998dc2d42183a434cc1dd19b919f4c740a009a","family_id":"qf-density14-circle-offset-residual-triple","seed":202610147025,"semantic_template":"qf-density14-circle-offset-residual-triple/[\"576\",[\"0\",\"576\",\"626\"]]"},"fields":{"title":"Squared Circle Distances, Residuals, and Boundary Membership","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, 1], r=24, ordered points=[[-1, 1], [23, 1], [24, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).","answer":{"kind":"expression","value":"([0, 576, 626], [-576, 0, 50], [inside, on-boundary, outside], [No, Yes, No])","explanation":"Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 576, 626], [-576, 0, 50], [inside, on-boundary, outside], [No, Yes, No])."},"skills":["Squared Circle Distances, Residuals, and Boundary Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0026","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc9b2ea9d1c1cfb30b16931c59aa5a5b8e2b8b44d2f91ab603c298c95df0a87f","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147026,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,0],0,\"collapsed\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 0], r=0, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([0, 0], [0, 0], 0, 0)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([0, 0], [0, 0], 0, 0)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0027","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69dde04405b48272c4b0e09e85378e933c725b6a231e5aab072f3b477ecfc33b","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147027,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,0],1,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 0], r=1, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([0, 1], [0, -1], 4, 2)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([0, 1], [0, -1], 4, 2)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0028","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d73599c6a2d013d9c7de533b8491ea32bd7c5a6243f4cde3773d6438f15b02","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147028,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,-10],2,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, -10], r=2, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([2, -10], [-2, -10], 16, 4)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([2, -10], [-2, -10], 16, 4)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0029","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f5b3421b59fc5881404b636fb705a37abe03fe534667251a8c922a97d13b234","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147029,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,-9],3,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, -9], r=3, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([1, -6], [1, -12], 36, 6)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, -6], [1, -12], 36, 6)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0030","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28ee168e771bd8411f8f3d4e7c26c034fea80552fc727f074a6911385c0e0c47","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147030,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,-8],4,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, -8], r=4, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([2, -8], [-6, -8], 64, 8)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([2, -8], [-6, -8], 64, 8)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0031","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e4bfc6b33815cc0c1e95d9c900b0357b59583e10ce47dc539907d11399ef8dd","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147031,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,-7],5,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, -7], r=5, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([-1, -2], [-1, -12], 100, 10)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, -2], [-1, -12], 100, 10)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0032","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1633fc7f80631c6524b8b3ff2aba15fc1b4ac602ccc5c54979e6257259fbfc52","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147032,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,-6],6,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, -6], r=6, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([6, -6], [-6, -6], 144, 12)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([6, -6], [-6, -6], 144, 12)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0033","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a6d483ddaad7a0c6beaf4217b72df758ab49f4c9489ce617871af166f5dad2e","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147033,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,-5],7,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, -5], r=7, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([1, 2], [1, -12], 196, 14)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 2], [1, -12], 196, 14)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0034","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5db34d08155f27e47af2f6c9f91fd4e90dde375c084eb3f3e28311fea940d29","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147034,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,-4],8,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, -4], r=8, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([6, -4], [-10, -4], 256, 16)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([6, -4], [-10, -4], 256, 16)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0035","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e78cf63a2063dc0babd2084ac4f2e658e71a5019f06e63758ead700acdd09a6b","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147035,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,-3],9,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, -3], r=9, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([-1, 6], [-1, -12], 324, 18)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, 6], [-1, -12], 324, 18)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0036","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfed8ac7da304b4c8902c3868702a98bbfa9a54e9fb852601dd95d8cdad05954","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147036,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,-2],10,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, -2], r=10, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([10, -2], [-10, -2], 400, 20)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([10, -2], [-10, -2], 400, 20)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0037","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b041d3f7a1200efae08d23ac5153311fa5b84c27ced923d61c8852bab9979e1f","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147037,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,-1],11,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, -1], r=11, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([1, 10], [1, -12], 484, 22)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 10], [1, -12], 484, 22)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0038","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c74c6f68dc24359f5415a7b04194dab867db1aed2a25c53b0e2d18709347e2ea","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147038,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,0],12,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, 0], r=12, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([10, 0], [-14, 0], 576, 24)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([10, 0], [-14, 0], 576, 24)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0039","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"183fe4ff21d46371860e454427a46832674c2856333e803fb83bb67a4d117d2c","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147039,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,1],13,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, 1], r=13, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([-1, 14], [-1, -12], 676, 26)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, 14], [-1, -12], 676, 26)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0040","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69bae3e20a023d41461ba2fd2cb332916e70e35b47841cf0f5bb1d6d01ad542a","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147040,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,2],14,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 2], r=14, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([14, 2], [-14, 2], 784, 28)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([14, 2], [-14, 2], 784, 28)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0041","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1ed29a6eb39f7cbb73698b81c7f9fed689521e0e3de182cdd1bfb757b8e59e6","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147041,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,3],15,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, 3], r=15, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([1, 18], [1, -12], 900, 30)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 18], [1, -12], 900, 30)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0042","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df763eb01eaa94e201ab838bba0ba9b51c1b4899521f1c3361adc63ba35369ac","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147042,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,4],16,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, 4], r=16, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([14, 4], [-18, 4], 1024, 32)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([14, 4], [-18, 4], 1024, 32)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0043","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8a4c9a196171d5c8b71f3dda46d539e38f17f560f5fdb363fe27a33493a6517","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147043,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,5],17,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, 5], r=17, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([-1, 22], [-1, -12], 1156, 34)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, 22], [-1, -12], 1156, 34)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0044","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1da9b542c5b955c65206eea0d9f1e2743b798f2823b194123bfa1f62f4c820e6","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147044,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,6],18,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 6], r=18, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([18, 6], [-18, 6], 1296, 36)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([18, 6], [-18, 6], 1296, 36)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0045","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dccba20825e57c1fa84c0473d45f6701ab4d625701f87746c7231d83b7d0d61b","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147045,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,7],19,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, 7], r=19, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([1, 26], [1, -12], 1444, 38)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 26], [1, -12], 1444, 38)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0046","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e44080e74f17b1784b450d8cf8094c7c808881c91144329eaf3847f66725bf33","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147046,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,8],20,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, 8], r=20, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([18, 8], [-22, 8], 1600, 40)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([18, 8], [-22, 8], 1600, 40)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0047","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7899f2d5893fab7ac9f6d6263e57d2b6380a846f8ad28789129edd53b0131f6e","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147047,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,9],21,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, 9], r=21, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([-1, 30], [-1, -12], 1764, 42)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, 30], [-1, -12], 1764, 42)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0048","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d10d13b2660e1e80120a2afba601b1562f3880526808b196b00f8d2f4b6aa87","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147048,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,10],22,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 10], r=22, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([22, 10], [-22, 10], 1936, 44)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([22, 10], [-22, 10], 1936, 44)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0049","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f640c8e3921e305ccac60dd92d418f6a75132ba52ec986dff9ce11d303917186","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147049,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,11],23,\"y\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, 11], r=23, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([1, 34], [1, -12], 2116, 46)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 34], [1, -12], 2116, 46)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0050","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f1f8a2843fe427349d76df09a5aa34b82247da51502bfbc634caa21e3ce27de","family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147050,"semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,12],24,\"x\"]"},"fields":{"title":"Axis Antipodes, Squared Separation, and Diameter","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, 12], r=24, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.","answer":{"kind":"expression","value":"([22, 12], [-26, 12], 2304, 48)","explanation":"Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([22, 12], [-26, 12], 2304, 48)."},"skills":["Axis Antipodes, Squared Separation, and Diameter"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0051","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9637253a7e901771274441e05d5014c87ff2e7cef76b539373031051a172954c","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147051,"semantic_template":"qf-density14-circle-radius-scale-witness/[\"zero_radius\"]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=0, scale k=5. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(0, 0, 0, 0, No)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (0, 0, 0, 0, No)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0052","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"031709eb37ae692eef9b73934ab28af6f8f3cc0f6ee2d21177198e51f79916d2","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147052,"semantic_template":"qf-density14-circle-radius-scale-witness/[1,0]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=1, scale k=0. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(0, 1, 0, 0, No)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (0, 1, 0, 0, No)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0053","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb7ad13e0fc5039cca1a23864b2b6fa6c95a14cb1fab6b1b502d51a2716133fb","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147053,"semantic_template":"qf-density14-circle-radius-scale-witness/[2,3]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=2, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(6, 4, 36, 12, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (6, 4, 36, 12, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0054","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"134402da800a67b966e815e202e8a1f3bb111da97e6708b07c2ce839bf857e9a","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147054,"semantic_template":"qf-density14-circle-radius-scale-witness/[3,1]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=3, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(3, 9, 9, 6, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (3, 9, 9, 6, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0055","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ffe4ea3e1184f66807eaf78e21d23369ec98f536a7959d7813daeaf56af5a4f","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147055,"semantic_template":"qf-density14-circle-radius-scale-witness/[4,2]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=4, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(8, 16, 64, 16, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (8, 16, 64, 16, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0056","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1720e013a72e379085c48a9992ca57a094e8c579ce3b6866661c223ea23f512","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147056,"semantic_template":"qf-density14-circle-radius-scale-witness/[5,3]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=5, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(15, 25, 225, 30, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (15, 25, 225, 30, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0057","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3411d142bd6e78c51e43003401d26023a4f9b02eb312ea9a0780e2b81d50f8c","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147057,"semantic_template":"qf-density14-circle-radius-scale-witness/[6,1]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=6, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(6, 36, 36, 12, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (6, 36, 36, 12, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0058","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"877902c719373cee1a2af43d0a72da9ff99f1d96aa44a54cc0f6b09670602d82","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147058,"semantic_template":"qf-density14-circle-radius-scale-witness/[7,2]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=7, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(14, 49, 196, 28, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (14, 49, 196, 28, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0059","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d455aa930eb10ac3d82be0321d3ee03a083b9de67cb0d82fa464cec577fd6cc4","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147059,"semantic_template":"qf-density14-circle-radius-scale-witness/[8,3]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=8, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(24, 64, 576, 48, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (24, 64, 576, 48, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0060","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f586fb9ce7c6832b2559bb6482dc6df4017c6dceeb843769fb11fe7c54fefe6","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147060,"semantic_template":"qf-density14-circle-radius-scale-witness/[9,1]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=9, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(9, 81, 81, 18, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (9, 81, 81, 18, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0061","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35f5747c3ceac068dc4672e5f7d265ad939e173a8405eb3311d6549facab54e8","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147061,"semantic_template":"qf-density14-circle-radius-scale-witness/[10,2]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=10, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(20, 100, 400, 40, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (20, 100, 400, 40, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0062","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1276371e3cb52ec752959811ab23374faa33c1f9fe4638044e296c4549412cb4","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147062,"semantic_template":"qf-density14-circle-radius-scale-witness/[11,3]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=11, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(33, 121, 1089, 66, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (33, 121, 1089, 66, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0063","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"464f7a1cf92550d1a88f0da1a5ce5debd6e5d9d5eeb01aa586ea6c1a022111ba","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147063,"semantic_template":"qf-density14-circle-radius-scale-witness/[12,1]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=12, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(12, 144, 144, 24, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (12, 144, 144, 24, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0064","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d0cd0ac1820d4a83901fb084291caf20ad47389a1f2b37bc68c16f1c1802766","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147064,"semantic_template":"qf-density14-circle-radius-scale-witness/[13,2]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=13, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(26, 169, 676, 52, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (26, 169, 676, 52, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0065","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"118510fbdb2110dc79026337bfe1ca682d624f0035407ae58ec4c848aa70a45d","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147065,"semantic_template":"qf-density14-circle-radius-scale-witness/[14,3]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=14, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(42, 196, 1764, 84, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (42, 196, 1764, 84, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0066","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29598cc4d21139b19b766e4fc54efb4f4cccd235011c127ac7d294149690d94c","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147066,"semantic_template":"qf-density14-circle-radius-scale-witness/[15,1]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=15, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(15, 225, 225, 30, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (15, 225, 225, 30, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0067","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1f3f752788b300d6baf6e7480161b8d555a09e3e2ebc88267512ed57dada029","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147067,"semantic_template":"qf-density14-circle-radius-scale-witness/[16,2]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=16, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(32, 256, 1024, 64, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (32, 256, 1024, 64, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0068","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8cdb1cc6724cefb5bc53ef1e7847a42186da6316e19621176a71877871e36fb","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147068,"semantic_template":"qf-density14-circle-radius-scale-witness/[17,3]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=17, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(51, 289, 2601, 102, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (51, 289, 2601, 102, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0069","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b5d6d170d8ae0bd5d15c6faad833de969983f1815bef25f1993decb11b1cc7d","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147069,"semantic_template":"qf-density14-circle-radius-scale-witness/[18,1]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=18, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(18, 324, 324, 36, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (18, 324, 324, 36, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0070","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"595c0599652186fbf7616846aeaf9833693eb2df8b205054948bc79c38945bde","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147070,"semantic_template":"qf-density14-circle-radius-scale-witness/[19,2]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=19, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(38, 361, 1444, 76, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (38, 361, 1444, 76, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0071","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3024750db5885776cd2b484b4111ba92c5fa4a0b4d7cfd1c05437c2847b2505d","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147071,"semantic_template":"qf-density14-circle-radius-scale-witness/[20,3]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=20, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(60, 400, 3600, 120, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (60, 400, 3600, 120, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0072","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bc6c8f5b2c0b31588694575af0fed29460e434fa41a2f8cda5317fbbd9af66e","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147072,"semantic_template":"qf-density14-circle-radius-scale-witness/[21,1]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=21, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(21, 441, 441, 42, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (21, 441, 441, 42, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0073","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af481aff2aa04402a6661209c605ba5292d8db63f8e9ee131a6a5d2f8c41bcf5","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147073,"semantic_template":"qf-density14-circle-radius-scale-witness/[22,2]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=22, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(44, 484, 1936, 88, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (44, 484, 1936, 88, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0074","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8d2b1b533aeb7e626dfc85d745f77a6ef25901dbfe26aa0983161f19989057c","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147074,"semantic_template":"qf-density14-circle-radius-scale-witness/[23,3]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=23, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(69, 529, 4761, 138, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (69, 529, 4761, 138, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0075","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d61673fa2648bafabb28493832e6540720851e39b3c3a06b215cc7b98300ea2","family_id":"qf-density14-circle-radius-scale-witness","seed":202610147075,"semantic_template":"qf-density14-circle-radius-scale-witness/[24,1]"},"fields":{"title":"Radius-Parameter Scaling and Positivity","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=24, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.","answer":{"kind":"expression","value":"(24, 576, 576, 48, Yes)","explanation":"Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (24, 576, 576, 48, Yes)."},"skills":["Radius-Parameter Scaling and Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0076","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ec8cdfbe5f808036ec7a0a02e0919f33e8390ecc81a72c076061c4a0eab02d5","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147076,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"0\",[\"0\",\"1\",\"1\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=0, horizontal displacement dx=0, ordered candidate y values=[0, 1, -1] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(0, [0, 1, 1], [Yes, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (0, [0, 1, 1], [Yes, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0077","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ea7c9f035deea3ca6fcc735db1539e44c78b35e1d1c0da64cf4954e174dd171","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147077,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"16\",[\"16\",\"16\",\"0\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=5, horizontal displacement dx=3, ordered candidate y values=[4, -4, 0] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(16, [0, 0, -16], [Yes, Yes, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (16, [0, 0, -16], [Yes, Yes, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0078","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac4908faf4e5b60f4108c3fe42249c6739070a31632e2a018871b4a3423ae072","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147078,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"-3\",[\"0\",\"1\",\"4\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=1, horizontal displacement dx=2, ordered candidate y values=[0, 1, 2] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(-3, [3, 4, 7], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (-3, [3, 4, 7], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0079","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d2cf1069d0f890f531d2d2cd7051b20b834de12480282640fb778119b39430b","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147079,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"15\",[\"0\",\"16\",\"25\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=4, horizontal displacement dx=-1, ordered candidate y values=[0, 4, 5] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(15, [-15, 1, 10], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (15, [-15, 1, 10], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0080","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e019c75a97dff4ac5b47f189d57f8cda2597e59d25e91d88be6126fdf0856df3","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147080,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"25\",[\"0\",\"25\",\"36\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=5, horizontal displacement dx=0, ordered candidate y values=[0, 5, 6] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(25, [-25, 0, 11], [No, Yes, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (25, [-25, 0, 11], [No, Yes, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0081","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61060fbe6cb1c189740c6e6a6c2a6b46ff0e272536961dc595d6135a20676251","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147081,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"35\",[\"0\",\"36\",\"49\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=6, horizontal displacement dx=1, ordered candidate y values=[0, 6, 7] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(35, [-35, 1, 14], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (35, [-35, 1, 14], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0082","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1ae86f0f62f77dbcb1ba6407140526d38e1bf664390a5f757860b2d75dd6b4e","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147082,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"48\",[\"0\",\"49\",\"64\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=7, horizontal displacement dx=-1, ordered candidate y values=[0, 7, 8] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(48, [-48, 1, 16], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (48, [-48, 1, 16], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0083","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c20e72bd8f09eacec7441bfca46b270cc78849f6421befd34d186d7182d1f2c6","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147083,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"64\",[\"0\",\"64\",\"81\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=8, horizontal displacement dx=0, ordered candidate y values=[0, 8, 9] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(64, [-64, 0, 17], [No, Yes, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (64, [-64, 0, 17], [No, Yes, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0084","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f4e3de47808fee3e4dbca2fc77eafee7f37878dd6e52101a7bea0b32777527c","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147084,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"80\",[\"0\",\"81\",\"100\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=9, horizontal displacement dx=1, ordered candidate y values=[0, 9, 10] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(80, [-80, 1, 20], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (80, [-80, 1, 20], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0085","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dffc4434692996b79510eaceae68fcc0bbc9f7e92bf64b0046e8a4765ae0079","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147085,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"99\",[\"0\",\"100\",\"121\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=10, horizontal displacement dx=-1, ordered candidate y values=[0, 10, 11] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(99, [-99, 1, 22], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (99, [-99, 1, 22], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0086","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c24532ccee2c3b508660948fac3025b0704ddd51927e8eb1b3f18d9f5fd12c86","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147086,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"121\",[\"0\",\"121\",\"144\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=11, horizontal displacement dx=0, ordered candidate y values=[0, 11, 12] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(121, [-121, 0, 23], [No, Yes, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (121, [-121, 0, 23], [No, Yes, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0087","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3889cd2d29b3712505fd5f9c4fe3c415ef1517a95f02414c8e226c6b7dd86bf","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147087,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"143\",[\"0\",\"144\",\"169\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=12, horizontal displacement dx=1, ordered candidate y values=[0, 12, 13] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(143, [-143, 1, 26], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (143, [-143, 1, 26], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0088","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7efac06ea0e97cdaf9f943ce1dc61e044cce5dbcdd43a01efbbfc040d3ad22a","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147088,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"168\",[\"0\",\"169\",\"196\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=13, horizontal displacement dx=-1, ordered candidate y values=[0, 13, 14] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(168, [-168, 1, 28], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (168, [-168, 1, 28], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0089","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f22ac32c034cfee0b87b5d43e147145dda3faded0f8b06c3d0e735eab434f835","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147089,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"196\",[\"0\",\"196\",\"225\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=14, horizontal displacement dx=0, ordered candidate y values=[0, 14, 15] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(196, [-196, 0, 29], [No, Yes, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (196, [-196, 0, 29], [No, Yes, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0090","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d03c8faf781a7b5eb79442270c14f742818c70782330f729313a15c0bd080a6","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147090,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"224\",[\"0\",\"225\",\"256\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=15, horizontal displacement dx=1, ordered candidate y values=[0, 15, 16] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(224, [-224, 1, 32], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (224, [-224, 1, 32], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0091","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"790f48b98d215bafe309e79dea859d7b49d3e47e3f1ca5d0dd67c205b87ea832","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147091,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"255\",[\"0\",\"256\",\"289\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=16, horizontal displacement dx=-1, ordered candidate y values=[0, 16, 17] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(255, [-255, 1, 34], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (255, [-255, 1, 34], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0092","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cd492dd71f1e296118706ff7e2e2d7e8f23173d8b99471116256faf168d5a61","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147092,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"289\",[\"0\",\"289\",\"324\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=17, horizontal displacement dx=0, ordered candidate y values=[0, 17, 18] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(289, [-289, 0, 35], [No, Yes, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (289, [-289, 0, 35], [No, Yes, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0093","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e09a4bf2802e53a603640147348887e472ba3bf7c3c7e6878a0b16eaa3e2464","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147093,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"323\",[\"0\",\"324\",\"361\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=18, horizontal displacement dx=1, ordered candidate y values=[0, 18, 19] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(323, [-323, 1, 38], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (323, [-323, 1, 38], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0094","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a63a93ecc3283b3c0a8cffd878fab1d22fa00afab7ae576379fb35a4f04d399b","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147094,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"360\",[\"0\",\"361\",\"400\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=19, horizontal displacement dx=-1, ordered candidate y values=[0, 19, 20] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(360, [-360, 1, 40], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (360, [-360, 1, 40], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0095","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"978c3ab0106ff443b3f56ec3bd0e10005c61a3e1f74936445453707c8b560023","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147095,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"400\",[\"0\",\"400\",\"441\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=20, horizontal displacement dx=0, ordered candidate y values=[0, 20, 21] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(400, [-400, 0, 41], [No, Yes, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (400, [-400, 0, 41], [No, Yes, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0096","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25568ff3a2c785a6865a5c3b6aa1fa23ccedca8e985c396ee18d7e2a3d5ba6e0","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147096,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"440\",[\"0\",\"441\",\"484\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=21, horizontal displacement dx=1, ordered candidate y values=[0, 21, 22] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(440, [-440, 1, 44], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (440, [-440, 1, 44], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0097","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"495fab2a5a19e83442d1e4e82284234599a2138a25103765663ba923c754cdeb","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147097,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"483\",[\"0\",\"484\",\"529\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=22, horizontal displacement dx=-1, ordered candidate y values=[0, 22, 23] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(483, [-483, 1, 46], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (483, [-483, 1, 46], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0098","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5753a6d6e0cd193ff823b52cc4abb72d1b16e05b7314466adcdbe2ef62c0b94","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147098,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"529\",[\"0\",\"529\",\"576\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=23, horizontal displacement dx=0, ordered candidate y values=[0, 23, 24] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(529, [-529, 0, 47], [No, Yes, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (529, [-529, 0, 47], [No, Yes, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0099","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c63bdada342dcfecf9d8e677bd03a5efb18432737b5387f11d53dc9bf7ec1749","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147099,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"575\",[\"0\",\"576\",\"625\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=24, horizontal displacement dx=1, ordered candidate y values=[0, 24, 25] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(575, [-575, 1, 50], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (575, [-575, 1, 50], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261010-advance-20261014-r07-circle-0100","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c4aa9c1958e377d0a39ac5cf914911bf1b8f647dda5c061d214a074309521f2","family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147100,"semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"624\",[\"0\",\"625\",\"676\"]]"},"fields":{"title":"Circle Squared Remainder at a Fixed Horizontal Offset","stem":"Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=25, horizontal displacement dx=-1, ordered candidate y values=[0, 25, 26] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).","answer":{"kind":"expression","value":"(624, [-624, 1, 52], [No, No, No])","explanation":"Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (624, [-624, 1, 52], [No, No, No])."},"skills":["Circle Squared Remainder at a Fixed Horizontal Offset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}