{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0001","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3207e3889c8ad0e9260e1f3d69cc96a07c85587d2d7c0274308f5cbf87740a4d","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091001,"semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":2}"},"fields":{"title":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(0, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (0, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0002","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"857023670102981a6a1355785a8013d5e304d0bc324d5b2fdc5a1625a6ba4e15","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091002,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(1, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (1, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0003","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34327e091402e6ae832bc8f827a28d984ea81e34ad211e43d4aa44f69371f27c","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091003,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0004","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adb5f0f379b42b48dc89a4257ac9f9b18c5952eec2b362ffe8ea8a76762677f2","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091004,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(3, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (3, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0005","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"873177369f4b59db6a06cf941e2c54716ff9f4b3ab50739df4d168d67bb0fae3","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091005,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(4, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (4, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0006","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36f794ee43e06745b4e093915e33d800b68e5acd21fc327e68ef0f2b81920fee","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091006,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0, 1], 2, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0, 1], 2, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0007","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10924586b0a4a56a9fcec292f16d65c48887e96a65e09206eac9fc3a2812b3d3","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091007,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0008","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c92b3eb1baab34d64363e228ec952c45c7ce581886f9d555034196a9a8f3c631","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091008,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0009","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88f4ee212cd3e81fd932a5e1befe3df99d4b09609c5660b57b3df6b7d1c0bffa","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091009,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0010","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c823fbc06ee2acf49ceba753b08133e6664d34a2e61dafdaac955d3162a5d38","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091010,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0011","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b97968fb2fb8e5f762a437bcb9bf76943430347607407554a4cca95e1273f152","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091011,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0, 1], 2, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0, 1], 2, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0012","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"216b1fbfc7f67dc81622be2e89c33068d6cec1f470d0be38070298ff8c2ae3f7","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091012,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(1, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (1, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0013","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdc6f44101bd8c14178f654063301a5e99bf897a349129dc4766b38580cd1b39","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091013,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(0, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (0, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0014","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3030465b3ae7ffbdc3a0395cfc0084bd8a78d893f1e4e56dc9f6246180e51bf","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091014,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4], 5, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4], 5, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0015","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bed7fa85ec28a7f4537bdce46e1eb607d8592692f82fd0edf66e2c9b69478eb","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091015,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4, 5], 6, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 5], 6, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0016","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba2bf2bff474180086f0c84cfc56b4db6c518168218107b51b0651c515c3a88b","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091016,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(0, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (0, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0017","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ae61701fa495f42feabda5404402dc29127660763b44ec4bc03b6ea712440ee","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091017,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(4, [0, 1, 2], 3, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (4, [0, 1, 2], 3, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0018","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"396885f0674f0ef4581d021366e2d3d15be421af44cff70e5d595b2957f97aff","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091018,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(3, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (3, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0019","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41ea292be8c22c4df866d269b9eed4b73a022e80260c7753bd052f48921655c7","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091019,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0020","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb751270638be76a87b844a9c51b8a49fadfdc11a72d0436fa314d48096fcde3","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091020,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(1, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (1, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0021","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b8f2152828dbe688f18511e2f7161a326431934ac54112ae4b80a5daf44dc10","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091021,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(2, [0, 1], 2, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0, 1], 2, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0022","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d549189c4928d3cd1a15ee1bd2489381530e5beab607bc5c7d46d69e67869ef","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091022,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(0, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (0, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0023","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10da0d4562be46ec4b47ecf24e17abc9b0703934f2cf6ad0c0f181ede4c289d9","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091023,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3], 4, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (6, [0, 1, 2, 3], 4, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0024","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac2e4d47693f3e433f0a0757522dd37eec20b06c2d66fcf5c282c0871056647d","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091024,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=5. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4], 5, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4], 5, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0025","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2138be32ee4a72c4899c97fd769bd1432f9428b56d0106cb8f2fe768f54928a7","family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091025,"semantic_template":"qf-mod40-stabilizer-stabilizer-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":"The Point Stabilizer Enumerates Every Fixing Group Element","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).","answer":{"kind":"expression","value":"(4, [0], 1, Yes, all_group_elements_checked)","explanation":"Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (4, [0], 1, Yes, all_group_elements_checked)."},"skills":["The Point Stabilizer Enumerates Every Fixing Group Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0026","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d85de8f9c5abd2c04a24ef2bd52b8d89fae201a248b32d26a5e3492e9467f4a","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091026,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":2}"},"fields":{"title":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0027","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"392e6fc6830314eee9f64863a53a691b359be3316e6f2e779138c1813b0eaa32","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091027,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0028","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f4b018f29c61f7e23337063b5725c77e1f2bed51ef3c00c897a330720091025","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091028,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0029","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dae6d9ea267c648e8b66d35a6048c09595296df40875e1bc2fdb3b1006be67e1","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091029,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0030","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ec8e6eaad21a53dba5a76e42da7ce6c5c1ead146491a5b649fae179040acf80","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091030,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0031","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c40217fa1482e3a96cc3991935f9c1a23396aec5566d5a6fc8232f2b71fd5d8","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091031,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":3}"},"fields":{"title":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0032","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bd3fde16ad8275a63b26e2724510b3b67a8c6fa2541854b1f304a607f4251a8","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091032,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0033","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c630e303dfc566ca183aabc6438184ddf82f1b7956d9a8c666a857ed10427897","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091033,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0034","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b1471aa9e7f2e7ea566fc75c8c5a94838cc9cb06492b09a96a4d3f418d0585f","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091034,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0035","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8cbc7f1c48fa52ddb7f58027450a3a3a58e2d7ee097009768274d0fd6ee7b95","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091035,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0036","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0b804ad163f897b01e5b3ca588cd8ebe287814ad7edb84a43c063947d0b04ca","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091036,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0037","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7690b415dd6f520dfe5250b94796c60c531d00595119e3879551eae00be9b130","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091037,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0038","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39cf9bf8f3476b1410397d4b31d7fcbb6f56a74c18829c40da107908713cc2ad","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091038,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0039","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85305a85b83181454ae305941e4c3dae27530d2be6f1a4a106110979469f0352","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091039,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0040","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89145ceb126e2f5e249bce37712e9d1d3982dfce390fe00a95647adb4a6fbbb4","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091040,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0041","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6394eb3fceabd1fdfd09b8aff555e35a2c4ea0526b41b64bb5f53e30ec93b67","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091041,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0042","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c2fe534777c0d0609fa705267e56f5dfd7fa8981791878f42b5f1e725a73b3","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091042,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0043","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11e961b2db1fd7a8db302ba8f51884779a4a86f35fd2655bb220022fb2739155","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091043,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0044","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8aea69016c879c7b7c931ff44b9f7ccb18c0e075c24c46ccd70d39ba7617d8f8","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091044,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0045","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4c25267e0900383a1283d3cae6ac16a2630a8143f298ea57c89b9a0d37d1d1a","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091045,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0046","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12f94f9b869f7ceb40c8653ffe5af57986c2508a231bf9da7b69e35a7fb07d31","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091046,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0047","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf64f7eec6cc57b01b3f2c1b8115be409d8464317a224b4e7b59ced677fb42c8","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091047,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0048","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e13a1369a44a265af29eff284b52371934149ed0ff89877abb5fdae8317fc5b","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091048,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2, 3], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0049","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2683be382273c5f6ab7bb5f87306c7e47d28fa6954fb33f3f6eb8fbf22e20576","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091049,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=5. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0050","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c5ad081db57363a9199a5778d45ff162910ba93d37e3b46f30896cccd0b4b1f","family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091050,"semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"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":"Identity, Product and Inverse Closure Certify the Full Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).","answer":{"kind":"expression","value":"([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)","explanation":"First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)."},"skills":["Identity, Product and Inverse Closure Certify the Full Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0051","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"824c5928b660166102a04a1f6f09a5f76aed11a1bdc68577ff24f0024977f170","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091051,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1],[1,0]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":2}"},"fields":{"title":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(0, 0, 0, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 0, 0, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0052","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3222f6491f9c8950ea6ec14b64325cfc518e046497121dcace711d656d67103a","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091052,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(1, 1, 2, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 1, 2, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0053","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d0e270998b5a52c068acf8a935ea454ec9822c38a3bd6fe6491b144025cfe5f","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091053,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(2, 2, 0, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (2, 2, 0, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0054","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"703eb7cd9e8cd6365d3f5052d7f24ea4f9333867b09e1b847c7ed162ae9e837d","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091054,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(3, 3, 1, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 3, 1, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0055","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b8fe35b4d40e46b3042f6f06c4d419600bc50a753ac3c560e96fb673e731beb","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091055,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Test group_element=4. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(4, 4, 2, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (4, 4, 2, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0056","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7e9f76a33f6a354e3ff51069069937fbf9124834f8887991c71dbb542df87a6","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091056,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(1, 2, 2, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 2, 2, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0057","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e32ca0326d4f06941e42c906654293ba0a7d2dc1a49a312c8962d64e2037237","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091057,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(0, 2, 2, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 2, 2, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0058","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f59e114b6448e376228593e8d7334c278c6d50c2e90877b2d61322d6f0f3bc5","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091058,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(3, 2, 1, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 2, 1, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0059","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"901fc1d724cff4f7b341efacc145d9658d0c08ab1206396f469a6733af88d12b","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091059,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(3, 2, 0, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 2, 0, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0060","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95d31b9447f79f8824c761ed743c7bcc9bb8cc7915136d9c0217bbce227a2d5f","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091060,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(3, 2, 5, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 2, 5, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0061","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6572b06623cc82ae93f5b8ef8f11244a25f82ed54564ca8d73f26b204a3fcd1","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091061,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(0, 2, 2, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 2, 2, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0062","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee8d612e26f407944791f4242b819e62504c886dd34b9ff078afc849ca07b3b8","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091062,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(2, 1, 0, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (2, 1, 0, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0063","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a13972e50837291d8fa4b1bef1e48aff4de62f98fd4fd71fd3a602b4d104036e","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091063,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(0, 0, 0, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 0, 0, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0064","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"384dc760af48bed0728a8c834f43c571eb24beda21311eac6f6669e56d821e2a","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091064,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(3, 6, 6, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 6, 6, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0065","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f201099b35e405a67367378de9b49234c0d9d8be306a99748457cb512d553b32","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091065,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(2, 6, 6, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (2, 6, 6, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0066","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d14cd062d063ce9a722c50ab9bab9c7e588bb8228c00ca0d47b5a11f910e5a72","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091066,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(1, 0, 1, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 0, 1, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0067","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c34ab18aa9840f5873806555ca5566ebf3dd810ce0a2032644040fb7583ae558","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091067,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(1, 4, 4, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 4, 4, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0068","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdc49569ca4b0d47acc7c8faaa7dcffaa281a16ffbcafd8aa64772a1f4212db3","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091068,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(1, 3, 0, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 3, 0, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0069","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34b9037e51a3e191339a10d932a5f81e3134103d9e3ad6d1d983eb22d4675093","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091069,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(3, 2, 0, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 2, 0, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0070","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5f2cdcfda06dd324490cf88cb92eab363127431f5e184e456fc025ac5008d1c","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091070,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(1, 1, 2, No, No)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 1, 2, No, No)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0071","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcfde6705230b391f261bf0b1c110b05082046b02fe1d8b8d0db3095151a485e","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091071,"semantic_template":"qf-mod40-stabilizer-membership-audit/{\"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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(0, 2, 2, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 2, 2, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0072","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6338f6b636790f3b23c9cd113ddf9d2b4397f090a112e6673b6baa614f5349fa","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091072,"semantic_template":"qf-mod40-stabilizer-membership-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_element\":0,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":0,\"point_count\":7}"},"fields":{"title":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(0, 0, 0, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 0, 0, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0073","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4161968203b7c3827b4fec75e88a05c66eceeee2d3a2434e5dba00b53f14f8fd","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091073,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(2, 6, 6, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (2, 6, 6, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0074","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab4db93bd7f3278e6b1cd5fd922ec1897b14708a8905d814e11a64278564ae15","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091074,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=5. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(3, 5, 5, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 5, 5, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0075","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52e9570b5fdd5d180d77e35243864b0fee4b8443985bff52e5550802d0c7eb09","family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091075,"semantic_template":"qf-mod40-stabilizer-membership-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_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":"Fixing This Point Is Exactly Membership in Its Stabilizer","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).","answer":{"kind":"expression","value":"(0, 4, 4, Yes, Yes)","explanation":"Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 4, 4, Yes, Yes)."},"skills":["Fixing This Point Is Exactly Membership in Its Stabilizer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0076","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1e257c74816d3dce0528d2f966a82b04025415e6fb161ec6d24641b01622000","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091076,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":1,\"point\":0,\"point_count\":2}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Compare other_point=1. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0077","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"227712293c50dd45d946ef73364d186ab5317f8d93d4135f8b73d41c8233e714","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091077,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"other_point\":2,\"point\":1,\"point_count\":3}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Compare other_point=2. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0078","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea3ed17e99357c70bea8c9ee6316266fecc22c1515659621649ce7e9b7d27980","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091078,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"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,\"other_point\":3,\"point\":2,\"point_count\":4}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0079","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5544dc4b0ac730942a7ab32be0ae6081e92de15e3047d1eaceb0dfbf2aa99fc6","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091079,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":4,\"point\":3,\"point_count\":5}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Compare other_point=4. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(3, [0], 4, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (3, [0], 4, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0080","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd23c0373f513962a2fc4aeb4e312a50e29c400f64283193161990027a503766","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091080,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":5,\"point\":4,\"point_count\":6}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Compare other_point=5. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(4, [0], 5, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (4, [0], 5, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0081","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"700aafabbb41e1e63f6ab9bb575991567988e6a0697485da4e75960c609dbe6d","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091081,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":0,\"point\":2,\"point_count\":3}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=0. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0, 1], 0, [0], No, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0, 1], 0, [0], No, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0082","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9de65240884baee7abecb4bc88779932857c0c9c12f6708ac343a059f391bc","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091082,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"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,\"other_point\":3,\"point\":2,\"point_count\":4}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0], 3, [0, 1, 2], No, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0, 1, 2], No, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0083","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cf329977a69a37900ab62cb3fb022a8cd46da5ab17efe5e3290b6cfc23b38ff","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091083,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"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,\"other_point\":3,\"point\":2,\"point_count\":5}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0084","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"490f6c6601dae2a7fb84c6b00b0a7c8f7a09b53474a564f246a13688a1d606ea","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091084,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":3,\"point\":2,\"point_count\":6}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0085","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ea4d42153607737d6a6d7e9b264958ed2c4f29f47f0f905881b06dad26c1e93","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091085,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":3,\"point\":2,\"point_count\":7}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0086","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38652df900d5d029f35ad459f587a94a00668febab2e47868debb34601ad5c06","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091086,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":4}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0, 1], 3, [0, 1], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0, 1], 3, [0, 1], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0087","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"494212dc1cb9ced95ad5c041c307ea074798ee5098945f96c9bc9767db887785","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091087,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"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,\"other_point\":2,\"point\":1,\"point_count\":5}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Compare other_point=2. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0088","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e519b920b057fc7d828d795706c846bee76a0023d9fe5264d016c0b20eaeb16","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091088,"semantic_template":"qf-mod40-stabilizer-point-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_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"other_point\":1,\"point\":0,\"point_count\":6}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Compare other_point=1. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0089","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4b7991bcb29b0c09533c3b9d6fefb2318f4b87bac40cbeed879153be93bf05a","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091089,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":0,\"point\":6,\"point_count\":7}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Compare other_point=0. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4], 0, [0], No, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4], 0, [0], No, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0090","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcd964b236e15778b910c0ef6433beb4dca5b2c5fd577b23c75bb8a6061d50b9","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091090,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":7,\"point\":6,\"point_count\":8}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Compare other_point=7. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4, 5], 7, [0, 1, 2, 3, 4, 5], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 5], 7, [0, 1, 2, 3, 4, 5], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0091","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24ebcd23a48220f4e19e61d620ebebc0b70499d34ad8264a3c9dd67ad20dc92e","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091091,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":1,\"point\":0,\"point_count\":5}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Compare other_point=1. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0092","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bf737e7099ea45a132732aeaebedba11a4c1b885800e297b02ddb89cff9925f","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091092,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"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,\"other_point\":5,\"point\":4,\"point_count\":6}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Compare other_point=5. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(4, [0, 1, 2], 5, [0, 1, 2], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (4, [0, 1, 2], 5, [0, 1, 2], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0093","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b347085489cf058a1f0aff0a20731197a9ce4377a994e06faa67f42a79316e23","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091093,"semantic_template":"qf-mod40-stabilizer-point-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_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"other_point\":4,\"point\":3,\"point_count\":7}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Compare other_point=4. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(3, [0], 4, [0, 1, 2, 3], No, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (3, [0], 4, [0, 1, 2, 3], No, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0094","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64e29102d55dcbfb5ea4ac5cc28aafd324358daa5bbbc73e2e6c1b1e7c5f85d2","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091094,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":3,\"point\":2,\"point_count\":8}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0095","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c9f4ea9f428fda6cafcfc0607d208c04eab32309ff459812a0e6a391e3417a9","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091095,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":2,\"point\":1,\"point_count\":9}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Compare other_point=2. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0096","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d7e99753c0a05260681cf55347793eed50a371875a6fd1aacd96fb579c71ff5","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091096,"semantic_template":"qf-mod40-stabilizer-point-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":6}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(2, [0, 1], 3, [0, 1], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0, 1], 3, [0, 1], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0097","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04f59d95ece03d50c8976bcff038d6ee58e1cf632f4636ab9d8eeda41f2f5772","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091097,"semantic_template":"qf-mod40-stabilizer-point-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_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"other_point\":1,\"point\":0,\"point_count\":7}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Compare other_point=1. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0098","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ee74e7ed518b82522b27b5d329b5e3a3768347d9e0733e7837423d742c4b93e","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091098,"semantic_template":"qf-mod40-stabilizer-point-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_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"other_point\":7,\"point\":6,\"point_count\":8}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","stem":"Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Compare other_point=7. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3], 7, [0, 1, 2, 3], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (6, [0, 1, 2, 3], 7, [0, 1, 2, 3], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0099","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07bbe7e7914373fba5f5d81764ea2853afb84d3e5238632ed7b71596c65a7fa4","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091099,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":6,\"point\":5,\"point_count\":9}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=5. Compare other_point=6. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4], 6, [0, 1, 2, 3, 4], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4], 6, [0, 1, 2, 3, 4], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261109-r01-finite-group-stabilizer-0100","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4b38648b7f43c1d4b6c8a40c87dc88b4640c07572f7bb8cc0cc9c650d0eb6f9","family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091100,"semantic_template":"qf-mod40-stabilizer-point-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_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,\"other_point\":5,\"point\":4,\"point_count\":10}"},"fields":{"title":"Point Stabilizers May Agree without Becoming Orbits or Kernels","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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Compare other_point=5. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).","answer":{"kind":"expression","value":"(4, [0], 5, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)","explanation":"Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (4, [0], 5, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)."},"skills":["Point Stabilizers May Agree without Becoming Orbits or Kernels"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}