{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0001","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fad62d5bb03decc223ab97d5c0b0043f698f65e4de6c288f23ef0947ffbecde","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081001,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":2}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(0, [0, 1], 2, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1], 2, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0002","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1e7466fad8726e0460b6c07639f562c1cf58a03b7d4e103d118b27a9e09d3a0","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081002,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":3}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=1. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(1, [0, 1, 2], 3, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (1, [0, 1, 2], 3, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0003","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d868425f0894fb668d4c2d8e32b6f94f0249b46046ff1329310efacd19e0723","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081003,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":4}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3], 4, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3], 4, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0004","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e78e3b2b2a8727df65d2b150362ed0a3f53ffddad5d053744873a81afd399e3b","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081004,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":3,\"point_count\":5}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=3. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0005","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d06b377a7f8de017936ad96459b01392bac05de4fef6b876a8a391869faf51f7","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081005,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":6}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=4. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0006","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3b70686b29bb72d0f8bebe224c971469d4636ec625f5f1061aa8d6c3dcf470b","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081006,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":3}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [2], 2, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [2], 2, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0007","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1802dab4f167d1f99676794628d41e77028040529c7141c8b2cbf8439771f759","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081007,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":2,\"point_count\":4}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 2], 3, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2], 3, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0008","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7e2432c9ebe3e879feda4556adf04d5674cac31b9a63ed73b2841c3b27e8e6e","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081008,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":5}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3], 4, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3], 4, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0009","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb4f802fb160d9ed6e471b8173db948e78f36fded64bacd0004e103d70eebe7","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081009,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":6}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0010","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f6f07ade3e824d2cddac1dc0b2e804cda1b853af4ab556bef733b30fe822ad2","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081010,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":2,\"point_count\":7}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0011","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d979aa3784e861b089fb640e3b8d759140fb706b35630340b247608ef1d92df","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081011,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":4}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [2], 2, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [2], 2, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0012","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71c256d12b8ebad1c42c3836ac17b6c888c0bca0e0e6aa9b377d876b55f3bf2e","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081012,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":5}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=1. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(1, [0, 1, 2], 3, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (1, [0, 1, 2], 3, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0013","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"554fb9054b98efc218a59622655641c9f8d4f3d57f9fc2d03aabe519f12aa3d1","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081013,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":0,\"point_count\":6}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3], 4, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1, 2, 3], 4, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0014","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c98a4ee3332d8a31c61d85c635b696914dcc67634930c284b6b2e9f030eebced","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081014,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":6,\"point_count\":7}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=6. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(6, [6], 5, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (6, [6], 5, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0015","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c67afe2afc89425b50cb85defc1893252c88dad8970d68f381d0446ee9f28e70","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081015,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":6,\"point_count\":8}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=6. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(6, [6], 6, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (6, [6], 6, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0016","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ca7f15d711a77395c7d6e3d13118489464768fe7680c7b0a84ba0f9fd609685","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081016,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":5}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(0, [0, 1], 2, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1], 2, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0017","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4d8739464827320dad873394933ab76ae1e69f4783d797e6c0256f5204535d1","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081017,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":4,\"point_count\":6}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=4. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(4, [4], 3, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (4, [4], 3, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0018","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cee8c558bc04bbb50973b18f892bef5a4f21338a65b32d7337294066f4e3d486","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081018,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":3,\"point_count\":7}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=3. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3], 4, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (3, [0, 1, 2, 3], 4, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0019","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0b9dae6b54ca9a0dd85d8761c01d0f5d7538877e7ee66412419499bd365fc54","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081019,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":8}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0020","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5212bcaf0693ec031b6fb65b9a3fca6491b3c9e3ee383f941c0bcfa0c93e4a2","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081020,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":1,\"point_count\":9}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=1. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0021","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfd5a05e76bab8761de0ee6ba5f6c36cc821d86d4c47d003808fe6ddafe0e116","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081021,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":6}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(2, [2], 2, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [2], 2, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0022","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74133fc8ea69de8cb893c94cd44daf1203c96d5e7f97ec80b9299f7d2280e876","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081022,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":0,\"point_count\":7}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 2], 3, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1, 2], 3, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0023","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faa2a0ffea5fcfc0e4c6a856ba43400aa0875aeaf03c2943a6c43c8a599b7fd1","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081023,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":6,\"point_count\":8}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=6. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(6, [6], 4, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (6, [6], 4, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0024","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9344518e0afb278442274546b0c1a9011964ff94ec00de96d06f11910d353a05","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081024,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":5,\"point_count\":9}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=5. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(5, [5], 5, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (5, [5], 5, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0025","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"915ae72659bf8d01f9acd89a08ebbbffbfe7eec6b40403f2a16f98d9d92bee80","family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081025,"semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":10}"},"fields":{"title":"An Orbit Uses All Group Elements, Not One Action Image","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(10)={0,…,9} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=4. Give (point, orbit, group_order, certificate).","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)","explanation":"Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)."},"skills":["An Orbit Uses All Group Elements, Not One Action Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0026","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecb0613655fa587f0a6416be8cb3d0867c06b61ca437c33044c73ca59d283f10","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081026,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1]], [0, 1], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1]], [0, 1], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0027","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb2c8d2818b191351379e9e923638fcaa4405066d7081cf7586917661124c8f6","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081027,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2]], [0, 1, 2], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2]], [0, 1, 2], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0028","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7832ad558cf0b6eaa3c4198f58a5aef28fe4a5ba07608ab4a5fd94fb2c79cfcc","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081028,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3]], [0, 1, 2, 3], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3]], [0, 1, 2, 3], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0029","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba7db8c2e520e7148eee9054ebc75a69ad53453865ed2f5431ec7527a1f638bd","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081029,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0030","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99fd19836c081a171e88f08ecb18dbc50b0a2cba071e5bfa4efb05307e5cd093","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081030,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5]], [0, 1, 2, 3, 4, 5], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5]], [0, 1, 2, 3, 4, 5], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0031","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96cf07991c31d1899cb400eb1a18043c70f4df7116df1758b5f9ab2b2d551452","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081031,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1], [2]], [0, 1, 2], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1], [2]], [0, 1, 2], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0032","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7236814abe949a17eb57360f5c7ca8b38f0567646ebaaff1ae1847f59ebfe7ee","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081032,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2], [3]], [0, 1, 2, 3], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2], [3]], [0, 1, 2, 3], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0033","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe331475b3722941a9812d947625e6b6686002c37c84c74c257902b2892ccab0","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081033,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3], [4]], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4]], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0034","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a2e6ab764a4d1961ab32061d2482e5183922ea75b90412d1017b020c400e01e","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081034,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0035","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05f83da649c4b7dcef89f75dc8e973ef397129a5e5c57dc41e79f8b6cc104e04","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081035,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0036","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6ec5e80dd95d877bf88118a61619376d9a0b359a623853d50269c042111817e","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081036,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1], [2], [3]], [0, 1, 2, 3], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1], [2], [3]], [0, 1, 2, 3], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0037","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7677c5b911d1ffa6720370ab1e84b9be71d3821a64e21a13039f62564d95512c","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081037,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2], [3], [4]], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2], [3], [4]], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0038","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fb2efde15c0e0aa77878247698c484892c04e49f894d0ad68f69e72c4782e9a","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081038,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0039","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74565e73800546cfe24bebc82cce972a514362fb174a7064c99832914f6beca3","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081039,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0040","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f69cdc186c1884feb7362e08a991218b7407b002a96534e2f3d03e5bed25bf3","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081040,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0041","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a89600fe8a8fecaf82d880b1161d49d3a81d9a707ecfa4e7efe54b7b5273d350","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081041,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1], [2], [3], [4]], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1], [2], [3], [4]], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0042","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15885d3abab14bba723ae90925ea6aa512ae6e84d72f01be1da6f4cc91617cab","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081042,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0043","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c10fcd588753a15603f10dac7813c2ec0ce455dc8f2ded3e75165043eb5246fc","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081043,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0044","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29cda3b0ba0a6d53b0eb1896575eb94d8dccdba6364585600fb8a95b47bd2205","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081044,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0045","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68c41fc42366a8eaa5e7d316dc2dc235a3241db2b5d24149965b4fa8666ddf2c","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081045,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0046","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75b68bcb5d3bd0e9063d21459677f3c2d02dd78240af01210e67d702a64c63bb","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081046,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0047","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23cd4955a7144725b7d7539ae096e4a941a724c5e8a4966a920d39574eb38f67","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081047,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0048","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3894b19f4fe4a9792229b9498b325b72f86c8f6d948d4dcaf2c34279062b52e","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081048,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0049","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91a78b4e7643c57cd40bfbee11c66bd8abdb02e216bbdaf2d14e219bda452d0c","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081049,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4], [5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0050","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea1237c752df48270145fca88905b56ec73303eb696493427326fadb9da94f69","family_id":"qf-mod39-orbit-orbit-partition","seed":202611081050,"semantic_template":"qf-mod39-orbit-orbit-partition/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10}"},"fields":{"title":"The Complete Orbits Partition the Whole Point Carrier","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(10)={0,…,9} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6], [7], [8], [9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes, Yes)","explanation":"Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6], [7], [8], [9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes, Yes)."},"skills":["The Complete Orbits Partition the Whole Point Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0051","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68d6c1ce41884f1aa60371feeb582a77deadb2f144a3d297545bb5cd1134731b","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081051,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"x\":0,\"y\":1}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=0, y=1. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(0, 1, [1], Yes, [0, 1], [0, 1], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (0, 1, [1], Yes, [0, 1], [0, 1], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0052","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4323631d530110c69abc8afc32c4f31663ae3754d3ae45d29fdbe788e34c54f0","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081052,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3,\"x\":1,\"y\":2}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=1, y=2. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(1, 2, [1], Yes, [0, 1, 2], [0, 1, 2], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (1, 2, [1], Yes, [0, 1, 2], [0, 1, 2], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0053","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5f303b93d9b6730475c9fe7294e7b66d3d35182474856ec3f5974c4ccaa7b71","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081053,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4,\"x\":2,\"y\":3}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 3, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0054","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24b2c03c1ee84453599a574d01ae9e3f0db348b73e8aa244801f9fb5547a2297","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081054,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5,\"x\":3,\"y\":4}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=3, y=4. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(3, 4, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (3, 4, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0055","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a765743593a73d3b7345915710f32f5b03af00cd96c2ee8fbaf33e5d1669f08","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081055,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6,\"x\":4,\"y\":5}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=4, y=5. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(4, 5, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (4, 5, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0056","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd91b5e0c2bbdc4042240454bfea9b58e4afa1147ecb29f952013e233411010c","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081056,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"x\":2,\"y\":0}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=0. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 0, [], No, [2], [0, 1], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 0, [], No, [2], [0, 1], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0057","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6fab3425c24195270e53d4ec7ae8c012b18963fb7713b6b1331bbd091e8205e","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081057,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4,\"x\":2,\"y\":3}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 3, [], No, [0, 1, 2], [3], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [], No, [0, 1, 2], [3], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0058","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc5c013969b67152820c6181b26e7e941805dc39166fda6906edb59a0d572a5e","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081058,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5,\"x\":2,\"y\":3}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 3, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0059","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27e6beb2685802813608e147ff1e012b3efd5467e9a30fede0f5f5614b49609a","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081059,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6,\"x\":2,\"y\":3}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 3, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0060","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ab580b3fbf88cad5176769fd2bf024ac4df4738b65389f97ee0feeecf4a3910","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081060,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7,\"x\":2,\"y\":3}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 3, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0061","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad6d6a434ad831d9da4cca3d0ba0e08adc2ecaa2732b19c87b9f83d68de8fe0e","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081061,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"x\":2,\"y\":3}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 3, [], No, [2], [3], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [], No, [2], [3], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0062","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9888e1700e6426577d77528a94232fdf0e7f1291a326c4185516f2859c81ed11","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081062,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5,\"x\":1,\"y\":2}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=1, y=2. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(1, 2, [1], Yes, [0, 1, 2], [0, 1, 2], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (1, 2, [1], Yes, [0, 1, 2], [0, 1, 2], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0063","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a68788e9e3fc6250291da93db432b601d703794e6acbcbea1e2148e7fe3ef3cf","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081063,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6,\"x\":0,\"y\":1}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=0, y=1. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(0, 1, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (0, 1, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0064","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b94da161cfca2b806a9bb531f761d5651383415a0a3c6410a243982fc2f0ecb","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081064,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7,\"x\":6,\"y\":0}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=6, y=0. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(6, 0, [], No, [6], [0, 1, 2, 3, 4], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (6, 0, [], No, [6], [0, 1, 2, 3, 4], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0065","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfd074aa494c510ad10788cb93ccaf2ac2f773bbc75bd84dcca1643929b0089e","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081065,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8,\"x\":6,\"y\":7}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=6, y=7. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(6, 7, [], No, [6], [7], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (6, 7, [], No, [6], [7], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0066","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18ac41cbb06691a7e9f4d268404841312f4c0a5f445a20e3daf9c0dea797e1b7","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081066,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"x\":0,\"y\":1}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=0, y=1. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(0, 1, [1], Yes, [0, 1], [0, 1], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (0, 1, [1], Yes, [0, 1], [0, 1], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0067","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7b099f74d39fd2001c79dc1f1b64634e00798388cff820ee61862fb1f095368","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081067,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6,\"x\":4,\"y\":5}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=4, y=5. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(4, 5, [], No, [4], [5], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (4, 5, [], No, [4], [5], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0068","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cc391505bf20cc420060a776dbe999fbf160135ca10d877d51af3574c452fba","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081068,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"x\":3,\"y\":4}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=3, y=4. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(3, 4, [], No, [0, 1, 2, 3], [4], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (3, 4, [], No, [0, 1, 2, 3], [4], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0069","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a73732d7bba541e2b4c4e36a8f333beb7fe6e985d290c4bbea49a004ccc2c32","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081069,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8,\"x\":2,\"y\":3}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 3, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0070","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ed153ebb0aa5b8bd522a4f20c82c9755e9c2776bd0eaade0a71d56bdc4b587a","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081070,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9,\"x\":1,\"y\":2}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=1, y=2. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(1, 2, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (1, 2, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0071","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acbab691c4edcb121158fb51011ef20e80272d6ac0a38edd06c7c9df9951dcc9","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081071,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6,\"x\":2,\"y\":3}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(2, 3, [], No, [2], [3], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [], No, [2], [3], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0072","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26e095f503a3d7db95b6fec9dfde88b4b30252c25a095162bbe9f89fd6593197","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081072,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7,\"x\":0,\"y\":1}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=0, y=1. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(0, 1, [1], Yes, [0, 1, 2], [0, 1, 2], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (0, 1, [1], Yes, [0, 1, 2], [0, 1, 2], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0073","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9da5a7c3cefb11ccc0fc91d21c72f70dae6bbc2cd5849af96f1e29c53b2477ae","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081073,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8,\"x\":6,\"y\":7}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=6, y=7. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(6, 7, [], No, [6], [7], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (6, 7, [], No, [6], [7], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0074","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"723d869ea1762dd4a4fbaad2f0c9ce72a17764ddc838cd94f6fd55e9aa6453d8","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081074,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9,\"x\":5,\"y\":6}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=5, y=6. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(5, 6, [], No, [5], [6], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (5, 6, [], No, [5], [6], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0075","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f41cd86ed2a027cdad7eee6bde98b70161df1f28296fe50fd97ef53a15d54ab3","family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081075,"semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10,\"x\":4,\"y\":5}"},"fields":{"title":"Existence of a Sending Group Element Agrees with Orbit Equality","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(10)={0,…,9} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=4, y=5. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).","answer":{"kind":"expression","value":"(4, 5, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)","explanation":"List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (4, 5, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)."},"skills":["Existence of a Sending Group Element Agrees with Orbit Equality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0076","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d5a35f9fdf6b1ecc5f8166b05315a743a82d71a0f3b57f84c9b3dd6cecc1a25","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081076,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1],[1,0]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":2}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=0. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, [0, 1], 2, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 0, 0, [0, 1], 2, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0077","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4cb6721e1cd5e46845a91bb4090bdc3fa8897a84e9ff2623354dfc84198aa78","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081077,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_element\":1,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":3}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=1. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(1, 1, 2, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 1, 2, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0078","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35cdefc44e831f2b9bb402b2c7f7d10e67d113ec614cc57fa38cb482f0e5ab6f","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081078,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_element\":2,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":4}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=2, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(2, 2, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (2, 2, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0079","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"120e62358f0f63ebe834ea7e17f95b3c0b9371141ac9f5f20cd39d7a0b2f1e47","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081079,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":3,\"point_count\":5}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=3. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(3, 3, 1, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 3, 1, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0080","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdacfa8c8e36403dfac6a77c79d532fcf62ab23fff0cb9f7458c2020d2d7d003","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081080,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_element\":4,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":6}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=4, point=4. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(4, 4, 2, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (4, 4, 2, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0081","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9cc37f7388753e11b46da50d81c1690eaae0eef861aada3762ef4c74433036d","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081081,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2],[1,0,2]],\"group_element\":1,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":3}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(1, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0082","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ed8120fe819d47fae26eda41d6647391ddade970efff91880e79df91ffb3d1d","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081082,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_element\":0,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":2,\"point_count\":4}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(0, 2, 2, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 2, 2, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0083","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f6e5ad855f582797d89228ccc20523b6d2fe3b1e0e64ac46fd45ed657c47f83","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081083,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_element\":3,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":5}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(3, 2, 1, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 2, 1, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0084","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94d33cd478c9b3dbe1c8de95d42e7040edecde194489cbe9a2fb940e1c88f4f0","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081084,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":6}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(3, 2, 0, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 2, 0, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0085","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb587f0c808874a35817f082e2b974659aace815f75e8c0398c117f33b2cde1c","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081085,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_element\":3,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":2,\"point_count\":7}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(3, 2, 5, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 2, 5, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0086","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f77907ed5901dabc0b9700683ece7ae413c508e39b01f9da70eeb78cf52acc6","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081086,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":4}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(0, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0087","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13e1818f6e136ca12f71ea1373209ef0b78c013000fef3cb3c7f9dd141855951","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081087,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_element\":2,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":5}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=2, point=1. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(2, 1, 0, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (2, 1, 0, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0088","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a24aa3c401adc8710724106e32f4e5a6ddb225c16326f12f8061543b9dfb5b12","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081088,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_element\":0,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":0,\"point_count\":6}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=0. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 0, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0089","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46f5c9792bd79a35a67446d71b53bfe70f7e4d6908a3b2752ee31012bf61c077","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081089,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":6,\"point_count\":7}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=6. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(3, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0090","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2960f5bc7e44a64c5bde62332ab727a9bdf3923f946b9edba5b365175e8d2188","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081090,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_element\":2,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":6,\"point_count\":8}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=2, point=6. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(2, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (2, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0091","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f15488cfc9ce56b4e64450f9f2b62e7979602def22c57904c7b52401d46d369","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081091,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_element\":1,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":5}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=0. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(1, 0, 1, [0, 1], 2, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 0, 1, [0, 1], 2, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0092","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ec95d14093bc1883ee7d0522afca5f37ba09a75748cc8ce1289569a40f65963","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081092,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_element\":1,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":4,\"point_count\":6}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=4. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(1, 4, 4, [4], 1, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 4, 4, [4], 1, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0093","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d5bb38bedd99eff0befbda6ff49ae3ace981de8dffb21a297178e40bad5f78c","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081093,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_element\":1,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":3,\"point_count\":7}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=3. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(1, 3, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 3, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0094","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25742c9ce43cca95fbb4efe8b7e49b336f6855358f52763135a7d43f16d60956","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081094,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":8}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(3, 2, 0, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 2, 0, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0095","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897f73b1f6b9289bc3ab703ec55c48feacb44bcbc2e4ae14c31ad227b8251720","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081095,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_element\":1,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":1,\"point_count\":9}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=1. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(1, 1, 2, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 1, 2, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0096","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9444101c38b80ba3800b0bd4b18eaa595d08b2cc2a1e02fe9fe4edbc8815fc3","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081096,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":6}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(0, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0097","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fe4127a64b95d2681143e6fc05ac5fffd6363b1e172c1bff827eaea7d02bd7c","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081097,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_element\":0,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":0,\"point_count\":7}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=0. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 0, 0, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0098","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcee903b48438b1a247f2214afa677c111d3d762b0efbe9ffcb62fd1b91cc0ff","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081098,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_element\":2,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":6,\"point_count\":8}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=2, point=6. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(2, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (2, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0099","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecddd10ecb7eecbcaeec8537ba8f180398c16ef9efb06e90864c05fcd64b1702","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081099,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":5,\"point_count\":9}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=5. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, 5, [5], 1, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 5, 5, [5], 1, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r01-finite-group-orbit-0100","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b0c34fbf9f9b7ab90a8485aadde339bffeaf4ae67526751bbabdd41380b25b2","family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081100,"semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_element\":0,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":10}"},"fields":{"title":"One Image Alone Is Not a Whole-Orbit Certificate","stem":"Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], declared identity=0, point carrier Fin(10)={0,…,9} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=4. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).","answer":{"kind":"expression","value":"(0, 4, 4, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit)","explanation":"Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 4, 4, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit)."},"skills":["One Image Alone Is Not a Whole-Orbit Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}