{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r07-quotient-vector-space-0001","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63231d4a13eb21a5e504d4dd637b4d1d7ea9d781c4a48be1c6295b56cc8994d1","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067001,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"u\":[4,4,3],\"v\":[2,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 1, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[2, 2, 3], u=[4, 4, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -2, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -2, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0002","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9147ba2c3a9e00061a0a5bf3ad90d37c0b76c816fb3ca198e37f06ef47262e07","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067002,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"u\":[5,6,3],\"v\":[3,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 2, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[3, 2, 3], u=[5, 6, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -4, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -4, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0003","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a8da2e5196589b5e72074bd1910b9eebf304a88fc2827142b309827b2bf281c","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067003,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"u\":[6,8,3],\"v\":[4,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 3, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[4, 2, 3], u=[6, 8, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -6, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -6, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0004","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e28cd274d0b298ff7bd84666c9b2c3bebd96f2f93eeba26d9d40a8d36be0a781","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067004,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"u\":[7,10,3],\"v\":[5,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 4, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[5, 2, 3], u=[7, 10, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -8, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -8, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0005","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69bd648736f2627af3e59b556259316cd81fe02bc3b56aed7182fb89aa7eed7d","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067005,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"u\":[8,12,3],\"v\":[6,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 5, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[6, 2, 3], u=[8, 12, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -10, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -10, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0006","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45a512136518b58f17dd1c3b518f12fe9bef56839063739904a915d30e083903","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067006,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"u\":[9,14,3],\"v\":[7,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 6, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[7, 2, 3], u=[9, 14, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -12, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -12, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0007","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44379e755dcbb48fcf499a31ccc578cef12ac1dee50c580b1c4e2ab2455c5434","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067007,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"u\":[10,16,3],\"v\":[8,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 7, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[8, 2, 3], u=[10, 16, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -14, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -14, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0008","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b761903c35f7782418276885cc268d330e4b7c53c3412cdce6cb60f7ea69354","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067008,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"u\":[11,18,3],\"v\":[9,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 8, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[9, 2, 3], u=[11, 18, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -16, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -16, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0009","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"522b696c117ea70e49faba2b7564797d770326e980bb8f5c6496ade85480b216","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067009,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"u\":[12,20,3],\"v\":[10,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 9, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[10, 2, 3], u=[12, 20, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -18, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -18, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0010","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d4a4cc3b171f0041ff4d59a9ffa214966bf4ba53826f93a03b9e48543c18ae8","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067010,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"u\":[13,22,3],\"v\":[11,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 10, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[11, 2, 3], u=[13, 22, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -20, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -20, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0011","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf35fe3f71fc97724f0fff2fac9a22bc6d36c8321b886c014895bacfce5ae468","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067011,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"u\":[14,24,3],\"v\":[12,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 11, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[12, 2, 3], u=[14, 24, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -22, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -22, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0012","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c74ec7017a0eca17b506ce32eecd592c48e76936e5d1120fb25f00bb2cad041a","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067012,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"u\":[15,26,3],\"v\":[13,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 12, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[13, 2, 3], u=[15, 26, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -24, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -24, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0013","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0281335e79cadec7fb81b94e1e06a7aa5367bf2533531ecc32c3de56ae441e4a","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067013,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"u\":[16,28,3],\"v\":[14,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 13, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[14, 2, 3], u=[16, 28, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -26, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -26, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0014","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8ba52daeb300e873f35f47b78760dd36ff2346490639b107edff3bf36c8e3d6","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067014,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"u\":[17,30,3],\"v\":[15,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 14, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[15, 2, 3], u=[17, 30, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -28, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -28, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0015","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eca8c5ba309c63eb842521eaaf1f646257e9429f3780ca2f015fe9e15b5110d2","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067015,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"u\":[18,32,3],\"v\":[16,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[16, 2, 3], u=[18, 32, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -30, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -30, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0016","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70c9c21918dbcc53acd72a517cd06887ff675c44b4f34a447ed96ebd69a73bd3","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067016,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"u\":[19,34,3],\"v\":[17,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 16, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[17, 2, 3], u=[19, 34, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -32, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -32, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0017","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701626b78f814e458ce5357da9055d086b231dbb3d8d76e8e698a0fd151d9334","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067017,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"u\":[20,36,3],\"v\":[18,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 17, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[18, 2, 3], u=[20, 36, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -34, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -34, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0018","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7146e36920146b36f3d94f514a99cd228b8e05c5551e26cf92f33ed38413292","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067018,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"u\":[21,38,3],\"v\":[19,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 18, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[19, 2, 3], u=[21, 38, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -36, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -36, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0019","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a79df9ee04507831356f4b610bfdc2d440cfd9ee90166dcc88dd3b623bad80a","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067019,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"u\":[22,40,3],\"v\":[20,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 19, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[20, 2, 3], u=[22, 40, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -38, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -38, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0020","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7820d65393b1b52187ebf511501620e4a0fba488739534e25251695d8eedb9de","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067020,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"u\":[23,42,3],\"v\":[21,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[21, 2, 3], u=[23, 42, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -40, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -40, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0021","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c732b9fe01015313af90605481d6a094a1c7268ea56fc172d5f71ffcac09758","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067021,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"u\":[24,44,3],\"v\":[22,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 21, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[22, 2, 3], u=[24, 44, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -42, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -42, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0022","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8efccc7783b7832f05776dcb88d54ada45f00caf50ffac3517e0ee4063e2c622","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067022,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"u\":[25,46,3],\"v\":[23,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 22, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[23, 2, 3], u=[25, 46, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -44, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -44, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0023","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d95696869acfc5bd51d6d75f14739d83dac914e92ed6aab80d50b926dbb473e","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067023,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"u\":[26,48,3],\"v\":[24,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 23, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[24, 2, 3], u=[26, 48, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -46, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -46, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0024","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4cb46f62feec35253331d8419daa540ea9bb674f592dd45bb9c82f493bbe2fd","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067024,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"u\":[27,50,3],\"v\":[25,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 24, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[25, 2, 3], u=[27, 50, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -48, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -48, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0025","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89e0ac8d20f2f5a5429803ad775efdfbb6ab79deacb03cfff79c4728a9ee4be7","family_id":"qf-mod37-quotient-coset-equivalence","seed":202611067025,"semantic_template":"qf-mod37-quotient-coset-equivalence/{\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"u\":[28,52,3],\"v\":[26,2,3]}"},"fields":{"title":"Equal Cosets Need Not Have Equal Representatives","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 25, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[26, 2, 3], u=[28, 52, 3]. Give (difference, equivalent, criterion, boundary).","answer":{"kind":"expression","value":"([-2, -50, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)","explanation":"Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -50, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)."},"skills":["Equal Cosets Need Not Have Equal Representatives"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0026","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1eff8d76be75eb7d3c271f9e1d15aff3a7ae745f45f6a18e1dee520362b9b975","family_id":"qf-mod37-quotient-coset-operations","seed":202611067026,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"scalar\":1,\"u\":[3,1,4],\"v\":[2,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 1, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[2, 2, 3], u=[3, 1, 4], scalar=1. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([5, 3, 7], [2, 2, 3], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([5, 3, 7], [2, 2, 3], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0027","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23405cc2f41d76bcc4651fa3e83d48e453afb21e0079438c196312c06a796738","family_id":"qf-mod37-quotient-coset-operations","seed":202611067027,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"scalar\":2,\"u\":[3,2,4],\"v\":[3,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 2, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[3, 2, 3], u=[3, 2, 4], scalar=2. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([6, 4, 7], [6, 4, 6], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([6, 4, 7], [6, 4, 6], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0028","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6496906aa05a2afacda429e95fae90994747725c38931daeedcb7aadb321d45c","family_id":"qf-mod37-quotient-coset-operations","seed":202611067028,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"scalar\":3,\"u\":[3,3,4],\"v\":[4,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 3, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[4, 2, 3], u=[3, 3, 4], scalar=3. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([7, 5, 7], [12, 6, 9], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([7, 5, 7], [12, 6, 9], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0029","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2be8a231c59089f5ee0742c3b2e5252fc433b2b63aab7cd2a29576f9024bf725","family_id":"qf-mod37-quotient-coset-operations","seed":202611067029,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"scalar\":4,\"u\":[3,4,4],\"v\":[5,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 4, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[5, 2, 3], u=[3, 4, 4], scalar=4. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([8, 6, 7], [20, 8, 12], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([8, 6, 7], [20, 8, 12], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0030","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b44a9850739452528c5408179020426f73009354115450379dc0edcebe6f5ef","family_id":"qf-mod37-quotient-coset-operations","seed":202611067030,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"scalar\":5,\"u\":[3,5,4],\"v\":[6,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 5, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[6, 2, 3], u=[3, 5, 4], scalar=5. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([9, 7, 7], [30, 10, 15], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([9, 7, 7], [30, 10, 15], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0031","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9011317e8aa24b187728375755055c959d62ca1a6372b2c04448867dc64ca45d","family_id":"qf-mod37-quotient-coset-operations","seed":202611067031,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"scalar\":6,\"u\":[3,6,4],\"v\":[7,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 6, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[7, 2, 3], u=[3, 6, 4], scalar=6. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([10, 8, 7], [42, 12, 18], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([10, 8, 7], [42, 12, 18], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0032","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5591743ae72b1aef34f46c65c3efae19bbf89511e19205bf480ecdca0b198d9","family_id":"qf-mod37-quotient-coset-operations","seed":202611067032,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"scalar\":7,\"u\":[3,7,4],\"v\":[8,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 7, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[8, 2, 3], u=[3, 7, 4], scalar=7. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([11, 9, 7], [56, 14, 21], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([11, 9, 7], [56, 14, 21], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0033","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cc8f4bb5b5b163ad6bded4e43fc5dd509a9a049260512702ec9be7d156d2f08","family_id":"qf-mod37-quotient-coset-operations","seed":202611067033,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"scalar\":8,\"u\":[3,8,4],\"v\":[9,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 8, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[9, 2, 3], u=[3, 8, 4], scalar=8. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([12, 10, 7], [72, 16, 24], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([12, 10, 7], [72, 16, 24], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0034","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70f8843805eaf73e5d188144c308e7b82275e4d9cd91263683b504da065fc420","family_id":"qf-mod37-quotient-coset-operations","seed":202611067034,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"scalar\":9,\"u\":[3,9,4],\"v\":[10,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 9, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[10, 2, 3], u=[3, 9, 4], scalar=9. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([13, 11, 7], [90, 18, 27], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([13, 11, 7], [90, 18, 27], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0035","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b14336c95882e5f4bc1108c3084a552bd52bd51a6b18428c40939591ac7036f","family_id":"qf-mod37-quotient-coset-operations","seed":202611067035,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"scalar\":10,\"u\":[3,10,4],\"v\":[11,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 10, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[11, 2, 3], u=[3, 10, 4], scalar=10. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([14, 12, 7], [110, 20, 30], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([14, 12, 7], [110, 20, 30], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0036","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e90eb1a14356b4a3b955ccdc44a1957357c5e66fc84bec31b9d0e89bf99c7695","family_id":"qf-mod37-quotient-coset-operations","seed":202611067036,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"scalar\":11,\"u\":[3,11,4],\"v\":[12,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 11, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[12, 2, 3], u=[3, 11, 4], scalar=11. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([15, 13, 7], [132, 22, 33], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([15, 13, 7], [132, 22, 33], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0037","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07e3821523f8bd696141d3ef9680cdb68a5ad69c35ed4e454c154bd75fd9e9c2","family_id":"qf-mod37-quotient-coset-operations","seed":202611067037,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"scalar\":12,\"u\":[3,12,4],\"v\":[13,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 12, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[13, 2, 3], u=[3, 12, 4], scalar=12. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([16, 14, 7], [156, 24, 36], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([16, 14, 7], [156, 24, 36], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0038","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"190efcc37d8c58a22df1dbd95c43537f77209f65a70da8455763652f0be21187","family_id":"qf-mod37-quotient-coset-operations","seed":202611067038,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"scalar\":13,\"u\":[3,13,4],\"v\":[14,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 13, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[14, 2, 3], u=[3, 13, 4], scalar=13. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([17, 15, 7], [182, 26, 39], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([17, 15, 7], [182, 26, 39], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0039","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"144caedda1be4a831fd05196e559db522ef06edd4bb3a426ccbd6fb7d118c377","family_id":"qf-mod37-quotient-coset-operations","seed":202611067039,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"scalar\":14,\"u\":[3,14,4],\"v\":[15,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 14, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[15, 2, 3], u=[3, 14, 4], scalar=14. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([18, 16, 7], [210, 28, 42], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([18, 16, 7], [210, 28, 42], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0040","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8408a618c20061f9b6704b140645c4f631967eb50a395f02b15b2c84452c90a","family_id":"qf-mod37-quotient-coset-operations","seed":202611067040,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"scalar\":15,\"u\":[3,15,4],\"v\":[16,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[16, 2, 3], u=[3, 15, 4], scalar=15. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([19, 17, 7], [240, 30, 45], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([19, 17, 7], [240, 30, 45], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0041","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f263fbf0d99636460237be1e5aa6416bd476d50bb678bc05d5362799f040c5c6","family_id":"qf-mod37-quotient-coset-operations","seed":202611067041,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"scalar\":16,\"u\":[3,16,4],\"v\":[17,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 16, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[17, 2, 3], u=[3, 16, 4], scalar=16. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([20, 18, 7], [272, 32, 48], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([20, 18, 7], [272, 32, 48], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0042","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"147f6e2041e39ea706a1a727e03102b0b4b52b2a8ece1da88d233ed3d476b4ea","family_id":"qf-mod37-quotient-coset-operations","seed":202611067042,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"scalar\":17,\"u\":[3,17,4],\"v\":[18,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 17, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[18, 2, 3], u=[3, 17, 4], scalar=17. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([21, 19, 7], [306, 34, 51], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([21, 19, 7], [306, 34, 51], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0043","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce287d7318be63e43cdfbf0bca2973c0c99ed4c61314126f5cb8aa9db495d6d0","family_id":"qf-mod37-quotient-coset-operations","seed":202611067043,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"scalar\":18,\"u\":[3,18,4],\"v\":[19,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 18, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[19, 2, 3], u=[3, 18, 4], scalar=18. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([22, 20, 7], [342, 36, 54], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([22, 20, 7], [342, 36, 54], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0044","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d98e7441f265761801be7d0881b1f9eace90cce4fa3d8701f9ae2fa2825ed70a","family_id":"qf-mod37-quotient-coset-operations","seed":202611067044,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"scalar\":19,\"u\":[3,19,4],\"v\":[20,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 19, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[20, 2, 3], u=[3, 19, 4], scalar=19. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([23, 21, 7], [380, 38, 57], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([23, 21, 7], [380, 38, 57], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0045","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84b73a14606521f1e9ffca8cad79f25fa128f4d6ee394a9b1aa77f2bfc0e5866","family_id":"qf-mod37-quotient-coset-operations","seed":202611067045,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"scalar\":20,\"u\":[3,20,4],\"v\":[21,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[21, 2, 3], u=[3, 20, 4], scalar=20. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([24, 22, 7], [420, 40, 60], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([24, 22, 7], [420, 40, 60], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0046","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da07e75587a7e821c84cbda4f651074a8acbff4d764834d22c591e2337b0b44b","family_id":"qf-mod37-quotient-coset-operations","seed":202611067046,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"scalar\":21,\"u\":[3,21,4],\"v\":[22,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 21, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[22, 2, 3], u=[3, 21, 4], scalar=21. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([25, 23, 7], [462, 42, 63], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([25, 23, 7], [462, 42, 63], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0047","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a1598d908e7b3a8bc9dcc6665ab206aebda7a966deb670a0a88e4c8ecd22f36","family_id":"qf-mod37-quotient-coset-operations","seed":202611067047,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"scalar\":22,\"u\":[3,22,4],\"v\":[23,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 22, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[23, 2, 3], u=[3, 22, 4], scalar=22. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([26, 24, 7], [506, 44, 66], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([26, 24, 7], [506, 44, 66], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0048","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d44707c216d98f1453cf546d78eb2f148383a19d1013758032f4fe115d9435f5","family_id":"qf-mod37-quotient-coset-operations","seed":202611067048,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"scalar\":23,\"u\":[3,23,4],\"v\":[24,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 23, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[24, 2, 3], u=[3, 23, 4], scalar=23. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([27, 25, 7], [552, 46, 69], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([27, 25, 7], [552, 46, 69], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0049","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a3a3afbbd6d163514160d6ab2712d4399d4b6fc787850ad147daad145c8059c","family_id":"qf-mod37-quotient-coset-operations","seed":202611067049,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"scalar\":24,\"u\":[3,24,4],\"v\":[25,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 24, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[25, 2, 3], u=[3, 24, 4], scalar=24. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([28, 26, 7], [600, 48, 72], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([28, 26, 7], [600, 48, 72], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0050","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfff077138e7044c3adf14a34a8905609b8afec752df799150b9860548b888a1","family_id":"qf-mod37-quotient-coset-operations","seed":202611067050,"semantic_template":"qf-mod37-quotient-coset-operations/{\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"scalar\":25,\"u\":[3,25,4],\"v\":[26,2,3]}"},"fields":{"title":"Report Representatives of Quotient Operations, Not Unique Coset Names","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 25, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[26, 2, 3], u=[3, 25, 4], scalar=25. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).","answer":{"kind":"expression","value":"([29, 27, 7], [650, 50, 75], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)","explanation":"Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([29, 27, 7], [650, 50, 75], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)."},"skills":["Report Representatives of Quotient Operations, Not Unique Coset Names"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0051","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07ef588bf6cce27d176408a9d8711d11886e2966f057cbdf6b7ebad2015b6997","family_id":"qf-mod37-quotient-representative-independence","seed":202611067051,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,1,0]],\"scalar\":1,\"shift_u_coefficients\":[-1],\"shift_v_coefficients\":[2],\"u\":[3,1,4],\"v\":[2,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 1, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[2, 2, 3], u=[3, 1, 4], shift_v_coefficients=[2], shift_u_coefficients=[-1] in supplied basis order, scalar=1. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0052","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80f929de09e1fd9ea6a4fd1525d3d7ea1a91f999d4f10b6b48ab2cffb3282afe","family_id":"qf-mod37-quotient-representative-independence","seed":202611067052,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,2,0]],\"scalar\":2,\"shift_u_coefficients\":[-2],\"shift_v_coefficients\":[3],\"u\":[3,2,4],\"v\":[3,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 2, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[3, 2, 3], u=[3, 2, 4], shift_v_coefficients=[3], shift_u_coefficients=[-2] in supplied basis order, scalar=2. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0053","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f7781b73d14ceba10d0d7a616253bfe3c3046706087c08374eb1c51a19cb036","family_id":"qf-mod37-quotient-representative-independence","seed":202611067053,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,3,0]],\"scalar\":3,\"shift_u_coefficients\":[-3],\"shift_v_coefficients\":[4],\"u\":[3,3,4],\"v\":[4,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 3, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[4, 2, 3], u=[3, 3, 4], shift_v_coefficients=[4], shift_u_coefficients=[-3] in supplied basis order, scalar=3. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0054","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98b4bba111ea84858d7a2257122ad790e1d41cfd3e072be73e4871a74d5d9111","family_id":"qf-mod37-quotient-representative-independence","seed":202611067054,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,4,0]],\"scalar\":4,\"shift_u_coefficients\":[-4],\"shift_v_coefficients\":[5],\"u\":[3,4,4],\"v\":[5,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 4, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[5, 2, 3], u=[3, 4, 4], shift_v_coefficients=[5], shift_u_coefficients=[-4] in supplied basis order, scalar=4. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0055","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a7c13570a76f57df8d1e54df1254cea59cc412712cd98417d2f980a5abe6e46","family_id":"qf-mod37-quotient-representative-independence","seed":202611067055,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,5,0]],\"scalar\":5,\"shift_u_coefficients\":[-5],\"shift_v_coefficients\":[6],\"u\":[3,5,4],\"v\":[6,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 5, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[6, 2, 3], u=[3, 5, 4], shift_v_coefficients=[6], shift_u_coefficients=[-5] in supplied basis order, scalar=5. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0056","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7d9eb7e4526c180016e444ce4d240a9dfdfaf153fb6e789269327ea4d116eb0","family_id":"qf-mod37-quotient-representative-independence","seed":202611067056,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,6,0]],\"scalar\":6,\"shift_u_coefficients\":[-6],\"shift_v_coefficients\":[7],\"u\":[3,6,4],\"v\":[7,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 6, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[7, 2, 3], u=[3, 6, 4], shift_v_coefficients=[7], shift_u_coefficients=[-6] in supplied basis order, scalar=6. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0057","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"109a053057421d9f1144406b9449eb583a9d6ea5cb2d15227b55be69bd1ee161","family_id":"qf-mod37-quotient-representative-independence","seed":202611067057,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,7,0]],\"scalar\":7,\"shift_u_coefficients\":[-7],\"shift_v_coefficients\":[8],\"u\":[3,7,4],\"v\":[8,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 7, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[8, 2, 3], u=[3, 7, 4], shift_v_coefficients=[8], shift_u_coefficients=[-7] in supplied basis order, scalar=7. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0058","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aec30a8a9c68f7e135092ad016ac8a963854d9763c15f7e717960e04df6ea3e9","family_id":"qf-mod37-quotient-representative-independence","seed":202611067058,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,8,0]],\"scalar\":8,\"shift_u_coefficients\":[-8],\"shift_v_coefficients\":[9],\"u\":[3,8,4],\"v\":[9,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 8, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[9, 2, 3], u=[3, 8, 4], shift_v_coefficients=[9], shift_u_coefficients=[-8] in supplied basis order, scalar=8. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0059","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f930e5c51cc041e2fe8929541422660845d1854e4ffe0bb6254929d833ff937d","family_id":"qf-mod37-quotient-representative-independence","seed":202611067059,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,9,0]],\"scalar\":9,\"shift_u_coefficients\":[-9],\"shift_v_coefficients\":[10],\"u\":[3,9,4],\"v\":[10,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 9, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[10, 2, 3], u=[3, 9, 4], shift_v_coefficients=[10], shift_u_coefficients=[-9] in supplied basis order, scalar=9. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0060","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2fa1deac4d0b41c07faf6b00d8ded0263c7e940d0b2dde57b5ad6665da6d244","family_id":"qf-mod37-quotient-representative-independence","seed":202611067060,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,10,0]],\"scalar\":10,\"shift_u_coefficients\":[-10],\"shift_v_coefficients\":[11],\"u\":[3,10,4],\"v\":[11,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 10, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[11, 2, 3], u=[3, 10, 4], shift_v_coefficients=[11], shift_u_coefficients=[-10] in supplied basis order, scalar=10. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0061","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d45b0aa50f01a1edb7cd8c748f02d10f6e2d53017b40f25a61a18783046cb43c","family_id":"qf-mod37-quotient-representative-independence","seed":202611067061,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,11,0]],\"scalar\":11,\"shift_u_coefficients\":[-11],\"shift_v_coefficients\":[12],\"u\":[3,11,4],\"v\":[12,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 11, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[12, 2, 3], u=[3, 11, 4], shift_v_coefficients=[12], shift_u_coefficients=[-11] in supplied basis order, scalar=11. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0062","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4983a9cb2312a179bce5a8f3e1ce60c726611e3e5caedb8acc4b96355922f240","family_id":"qf-mod37-quotient-representative-independence","seed":202611067062,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,12,0]],\"scalar\":12,\"shift_u_coefficients\":[-12],\"shift_v_coefficients\":[13],\"u\":[3,12,4],\"v\":[13,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 12, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[13, 2, 3], u=[3, 12, 4], shift_v_coefficients=[13], shift_u_coefficients=[-12] in supplied basis order, scalar=12. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0063","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e1881cf26dff4b7563273ea2b31344c61c30c26a48dc38eb1cc5d98dab6c26b","family_id":"qf-mod37-quotient-representative-independence","seed":202611067063,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,13,0]],\"scalar\":13,\"shift_u_coefficients\":[-13],\"shift_v_coefficients\":[14],\"u\":[3,13,4],\"v\":[14,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 13, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[14, 2, 3], u=[3, 13, 4], shift_v_coefficients=[14], shift_u_coefficients=[-13] in supplied basis order, scalar=13. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0064","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c57dca8b2c1b94481b15c718d4de7f1f395f80bfdda3de608542a1380974011c","family_id":"qf-mod37-quotient-representative-independence","seed":202611067064,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,14,0]],\"scalar\":14,\"shift_u_coefficients\":[-14],\"shift_v_coefficients\":[15],\"u\":[3,14,4],\"v\":[15,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 14, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[15, 2, 3], u=[3, 14, 4], shift_v_coefficients=[15], shift_u_coefficients=[-14] in supplied basis order, scalar=14. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0065","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9827877d9ac84715e3de9ba8757e47c16c26d2ae417aa4a521edd9fd17a9e20","family_id":"qf-mod37-quotient-representative-independence","seed":202611067065,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,15,0]],\"scalar\":15,\"shift_u_coefficients\":[-15],\"shift_v_coefficients\":[16],\"u\":[3,15,4],\"v\":[16,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[16, 2, 3], u=[3, 15, 4], shift_v_coefficients=[16], shift_u_coefficients=[-15] in supplied basis order, scalar=15. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0066","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a2602364753650db10f47bf37069f685f17398dcad6f4c4e9c2c35f5e388cd9","family_id":"qf-mod37-quotient-representative-independence","seed":202611067066,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,16,0]],\"scalar\":16,\"shift_u_coefficients\":[-16],\"shift_v_coefficients\":[17],\"u\":[3,16,4],\"v\":[17,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 16, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[17, 2, 3], u=[3, 16, 4], shift_v_coefficients=[17], shift_u_coefficients=[-16] in supplied basis order, scalar=16. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0067","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6066e64b0eda3ecb44cd45fc8c86f394f9b976b6ff22a34a549c055855b1be0f","family_id":"qf-mod37-quotient-representative-independence","seed":202611067067,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,17,0]],\"scalar\":17,\"shift_u_coefficients\":[-17],\"shift_v_coefficients\":[18],\"u\":[3,17,4],\"v\":[18,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 17, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[18, 2, 3], u=[3, 17, 4], shift_v_coefficients=[18], shift_u_coefficients=[-17] in supplied basis order, scalar=17. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0068","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cce27fb89a3ac1a7d55196f1832bf0da86d09a1df65e75e24048f70c68cbd985","family_id":"qf-mod37-quotient-representative-independence","seed":202611067068,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,18,0]],\"scalar\":18,\"shift_u_coefficients\":[-18],\"shift_v_coefficients\":[19],\"u\":[3,18,4],\"v\":[19,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 18, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[19, 2, 3], u=[3, 18, 4], shift_v_coefficients=[19], shift_u_coefficients=[-18] in supplied basis order, scalar=18. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0069","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53fa0373fc36369f7108469ee1c1ab52934be349584a75666ab36f256634a5d0","family_id":"qf-mod37-quotient-representative-independence","seed":202611067069,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,19,0]],\"scalar\":19,\"shift_u_coefficients\":[-19],\"shift_v_coefficients\":[20],\"u\":[3,19,4],\"v\":[20,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 19, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[20, 2, 3], u=[3, 19, 4], shift_v_coefficients=[20], shift_u_coefficients=[-19] in supplied basis order, scalar=19. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0070","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45d68f9e844584176a9ae625a8576967dd69b508d8595d0f4b41c4339fca2da2","family_id":"qf-mod37-quotient-representative-independence","seed":202611067070,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,20,0]],\"scalar\":20,\"shift_u_coefficients\":[-20],\"shift_v_coefficients\":[21],\"u\":[3,20,4],\"v\":[21,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[21, 2, 3], u=[3, 20, 4], shift_v_coefficients=[21], shift_u_coefficients=[-20] in supplied basis order, scalar=20. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0071","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5d2a4c25439ccc42bfb3091d339f98274decee38f0042b056cd2014f648d8b6","family_id":"qf-mod37-quotient-representative-independence","seed":202611067071,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]],\"scalar\":21,\"shift_u_coefficients\":[-21],\"shift_v_coefficients\":[22],\"u\":[3,21,4],\"v\":[22,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 21, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[22, 2, 3], u=[3, 21, 4], shift_v_coefficients=[22], shift_u_coefficients=[-21] in supplied basis order, scalar=21. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0072","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65ba58ccfbc0d3daf91d48c1eb0e92f12b49c1c11095e0fbf56155887c40bec4","family_id":"qf-mod37-quotient-representative-independence","seed":202611067072,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,22,0]],\"scalar\":22,\"shift_u_coefficients\":[-22],\"shift_v_coefficients\":[23],\"u\":[3,22,4],\"v\":[23,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 22, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[23, 2, 3], u=[3, 22, 4], shift_v_coefficients=[23], shift_u_coefficients=[-22] in supplied basis order, scalar=22. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0073","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1de10cc9e873d22986fe27cf97a787b38f8c2a24d28e0209b36e668ac8fb808b","family_id":"qf-mod37-quotient-representative-independence","seed":202611067073,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,23,0]],\"scalar\":23,\"shift_u_coefficients\":[-23],\"shift_v_coefficients\":[24],\"u\":[3,23,4],\"v\":[24,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 23, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[24, 2, 3], u=[3, 23, 4], shift_v_coefficients=[24], shift_u_coefficients=[-23] in supplied basis order, scalar=23. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0074","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d95fc64c9f2addac501842ec131399943c5f9b987d50d703aa4f24808700ea91","family_id":"qf-mod37-quotient-representative-independence","seed":202611067074,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,24,0]],\"scalar\":24,\"shift_u_coefficients\":[-24],\"shift_v_coefficients\":[25],\"u\":[3,24,4],\"v\":[25,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 24, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[25, 2, 3], u=[3, 24, 4], shift_v_coefficients=[25], shift_u_coefficients=[-24] in supplied basis order, scalar=24. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0075","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"832efabcd18a82cc3fba30c7ccf084d3b0f4448106e05dcee07a4386b83c8598","family_id":"qf-mod37-quotient-representative-independence","seed":202611067075,"semantic_template":"qf-mod37-quotient-representative-independence/{\"ambient_dimension\":3,\"basis_w\":[[1,25,0]],\"scalar\":25,\"shift_u_coefficients\":[-25],\"shift_v_coefficients\":[26],\"u\":[3,25,4],\"v\":[26,2,3]}"},"fields":{"title":"Changing Representatives by W Must Preserve Both Operations","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 25, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[26, 2, 3], u=[3, 25, 4], shift_v_coefficients=[26], shift_u_coefficients=[-25] in supplied basis order, scalar=25. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)","explanation":"Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)."},"skills":["Changing Representatives by W Must Preserve Both Operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0076","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f1e3313f139886e54f52a149fdd8ce3ae68f03b9826bf4eb4789e79552c67dd","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067076,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,1,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 1, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0077","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77b09299f277ec1d987dcd2b179772a07f7e8ab2923aab4a967840108409ead8","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067077,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,2,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 2, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0078","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f679761d83c98842aafce8068ab022589b411f0a27f013dd10a60b2ab3e55129","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067078,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,3,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 3, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0079","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdaddc13afaa0552391a4fade163334c2251e7ca1c78bb88f66f2b116923d97c","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067079,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,4,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 4, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0080","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9160e553c5d5a792fa95cabebe0a0e39694731f2d15a62366c58085d67558016","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067080,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,5,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 5, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0081","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb46f1c3fdb95a47db8c616b8384cd12119e38ce337380d1bdb48f1fec68b88","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067081,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,6,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 6, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0082","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6d1376d21aad81761cbc1a7a69211cfcc63e0b092817c9b5965a763cf383ced","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067082,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,7,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 7, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0083","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c88d174c7c2bcf54f38ca5094f44da2a4d2a2631bb0c8e7f1ddb2ab3622d5abd","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067083,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,8,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 8, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0084","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"731bc7f0b94a07bd1c55218026f1a8e758cb02092a734036f84b4ce186f9969b","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067084,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,9,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 9, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0085","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7259a6865e404a8ed31803f4baf6fec247ee989c313e24f23e86d28049621431","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067085,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,10,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 10, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0086","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fe743b64d3d5a0816a8a7d5a9397e13ab58ce2facbe96e263f1472427872267","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067086,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,11,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 11, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0087","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d713d70423dea431ac98ef06b48015ca74933dcf659f3a055a9976a870e57fe","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067087,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,12,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 12, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0088","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6f14c681527364936665e93051670977adea7e1b5a570919d6ec0347b2117e2","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067088,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,13,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 13, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0089","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b37d1ba48040b79037d92bba4248b7042ff536d01b2fd54d2d43450b7c8451b","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067089,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,14,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 14, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0090","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e01ec414304dca46f6cddd00ec736d151a1ee6311045df98fcda54dfff539be7","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067090,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,15,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0091","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d05bfa05eb69b130c766ff1238cb18f4bd941bb25d4f80da5d7d7bd94e166fb","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067091,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,16,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 16, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0092","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f507d7f69c7b50ecaf8d2dce9c9e9dfa7352f56ca23511760b23471d32f891a","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067092,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,17,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 17, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0093","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2007aec1b050fb6086ee4ef0ad6e1722bf2ef60b646de0df905f876364a34b3","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067093,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,18,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 18, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0094","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a427140b556e27226b99411c6c97d5e55881dec97ccec534b1d198875b368332","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067094,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,19,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 19, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0095","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0a7405dadb6f5e339cbb7b48cb18b5afe2d9c5636c29d911594ac1d5900123e","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067095,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,20,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0096","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae58d87e62062892976c2a07fee44140f0cedfc7508717cf2dd2e90418fadc1d","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067096,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,21,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 21, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0097","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49f7eef8e8f900b72eb704a645eb7737f27b99450c8417b51aa60fa5783cb4aa","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067097,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,22,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 22, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0098","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1532b1d24a3751f7708017688007be452085633387a99e37dd84c0f22a7a11ff","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067098,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,23,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 23, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0099","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cca2785e9fda65f9dd085de2455c05e8d4b6077f41817bea08c4c53025bdf8a8","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067099,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,24,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 24, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r07-quotient-vector-space-0100","batch_id":"batch-20261014-advance-20261106-r07-quotient-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e806cd569c72d0a95fecde9c39f91ad3ad741d308cafa65341c9837932410d50","family_id":"qf-mod37-quotient-dimension-formula","seed":202611067100,"semantic_template":"qf-mod37-quotient-dimension-formula/{\"ambient_dimension\":3,\"basis_w\":[[1,25,0]]}"},"fields":{"title":"Finite Quotient Dimension Subtracts Subspace Dimension","stem":"Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 25, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).","answer":{"kind":"expression","value":"(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)","explanation":"Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)."},"skills":["Finite Quotient Dimension Subtracts Subspace Dimension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}