{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0001","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42f945882e933f1245b76e59e9089eddd84f477b562048e4422aa6f7a5a6c12c","family_id":"qf-str22-vectorspace-addition","seed":202610224001,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-12,-3,2],\"right\":[-2,-8,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-12, -3, 2], right=[-2, -8, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-12, -3, 2], [-2, -8, 0], [-14, -11, 2])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -3, 2], [-2, -8, 0], [-14, -11, 2])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0002","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c6ad2e356f278e2f35675eeec239bad6e95a613e7797b5ba80d2cbace814989","family_id":"qf-str22-vectorspace-addition","seed":202610224002,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-11,-2,1],\"right\":[-1,-7,1]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-11, -2, 1], right=[-1, -7, 1]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-11, -2, 1], [-1, -7, 1], [-12, -9, 2])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -2, 1], [-1, -7, 1], [-12, -9, 2])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0003","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e6e45eef959b0cf4ef88672e028396e48633f3b1e02f75b54d843c5dc30d87f","family_id":"qf-str22-vectorspace-addition","seed":202610224003,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-10,-1,0],\"right\":[0,-6,2]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-10, -1, 0], right=[0, -6, 2]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-10, -1, 0], [0, -6, 2], [-10, -7, 2])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, -1, 0], [0, -6, 2], [-10, -7, 2])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0004","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72eca13cca33a04fed1b2f9d8a750a5f8cabf467df7928e92cf1c744ae3d5596","family_id":"qf-str22-vectorspace-addition","seed":202610224004,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-9,0,-1],\"right\":[1,-5,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-9, 0, -1], right=[1, -5, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-9, 0, -1], [1, -5, 0], [-8, -5, -1])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 0, -1], [1, -5, 0], [-8, -5, -1])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0005","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b9c09d6657bdf182780f89a44077dd5e1c27628134f647a14e8b8e305cc5403","family_id":"qf-str22-vectorspace-addition","seed":202610224005,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-8,1,-2],\"right\":[2,-4,1]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-8, 1, -2], right=[2, -4, 1]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-8, 1, -2], [2, -4, 1], [-6, -3, -1])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 1, -2], [2, -4, 1], [-6, -3, -1])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0006","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"033aa095fb007ffe550f9f3f3bac0598c3120906e86ad2ab2112ee8d82d150e8","family_id":"qf-str22-vectorspace-addition","seed":202610224006,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-7,2,-3],\"right\":[-2,-3,2]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-7, 2, -3], right=[-2, -3, 2]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-7, 2, -3], [-2, -3, 2], [-9, -1, -1])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, 2, -3], [-2, -3, 2], [-9, -1, -1])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0007","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb6ed9bc773301a1c8137de9899c67f993422be0b055531e94698bb681bc79d8","family_id":"qf-str22-vectorspace-addition","seed":202610224007,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-6,3,-4],\"right\":[-1,-2,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-6, 3, -4], right=[-1, -2, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-6, 3, -4], [-1, -2, 0], [-7, 1, -4])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, 3, -4], [-1, -2, 0], [-7, 1, -4])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0008","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e0e3e226656ca6b76ecf99b5e7a2f023162625b852be92f10d2d438aba8f885","family_id":"qf-str22-vectorspace-addition","seed":202610224008,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-5,-3,-5],\"right\":[0,-1,1]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-5, -3, -5], right=[0, -1, 1]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-5, -3, -5], [0, -1, 1], [-5, -4, -4])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, -3, -5], [0, -1, 1], [-5, -4, -4])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0009","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1849574ac5023a88a619acd111e0e83bf688c6169cc3d009f747a0245dc4d8f1","family_id":"qf-str22-vectorspace-addition","seed":202610224009,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-4,-2,-6],\"right\":[1,0,2]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-4, -2, -6], right=[1, 0, 2]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-4, -2, -6], [1, 0, 2], [-3, -2, -4])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, -2, -6], [1, 0, 2], [-3, -2, -4])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0010","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12e2fdaf4e33ffcf985ff437415eedcafbbf326ce949c4bd328d1409399425e9","family_id":"qf-str22-vectorspace-addition","seed":202610224010,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-3,-1,-7],\"right\":[2,1,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-3, -1, -7], right=[2, 1, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-3, -1, -7], [2, 1, 0], [-1, 0, -7])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, -1, -7], [2, 1, 0], [-1, 0, -7])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0011","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22de3ab000ecb533003086f926e329bd9570645f654af5b7a89c11195e2fdd48","family_id":"qf-str22-vectorspace-addition","seed":202610224011,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-2,0,-8],\"right\":[-2,2,1]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-2, 0, -8], right=[-2, 2, 1]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-2, 0, -8], [-2, 2, 1], [-4, 2, -7])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, 0, -8], [-2, 2, 1], [-4, 2, -7])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0012","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f8d82c08fa749358c250db258a0f9e1edcb013f0cda045ebf8c9355ef3acdad","family_id":"qf-str22-vectorspace-addition","seed":202610224012,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[-1,1,-9],\"right\":[-1,3,2]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-1, 1, -9], right=[-1, 3, 2]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([-1, 1, -9], [-1, 3, 2], [-2, 4, -7])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, 1, -9], [-1, 3, 2], [-2, 4, -7])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0013","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e311189193b5cc8f0cb02285fbd56dc40ec7e61958e95005b43b3793ce496fa","family_id":"qf-str22-vectorspace-addition","seed":202610224013,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[0,2,-10],\"right\":[0,4,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[0, 2, -10], right=[0, 4, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([0, 2, -10], [0, 4, 0], [0, 6, -10])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 2, -10], [0, 4, 0], [0, 6, -10])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0014","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0d38ea85108af22cbbdb39e3fc434cd2dd97853ffe8301f206f33dd65958b67","family_id":"qf-str22-vectorspace-addition","seed":202610224014,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[1,3,-11],\"right\":[1,5,1]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[1, 3, -11], right=[1, 5, 1]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([1, 3, -11], [1, 5, 1], [2, 8, -10])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 3, -11], [1, 5, 1], [2, 8, -10])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0015","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3dc1a1f1b20b5e227fe9a93c7dd8c22f8cd2e33be3b837ff0d8e0c390065aa4","family_id":"qf-str22-vectorspace-addition","seed":202610224015,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[2,-3,-12],\"right\":[2,6,2]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[2, -3, -12], right=[2, 6, 2]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([2, -3, -12], [2, 6, 2], [4, 3, -10])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, -3, -12], [2, 6, 2], [4, 3, -10])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0016","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f94302c96c856c3e111bd9922837f5b4b8c20b962843db9dd0446e1a630b7f9","family_id":"qf-str22-vectorspace-addition","seed":202610224016,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[3,-2,-13],\"right\":[-2,7,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[3, -2, -13], right=[-2, 7, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([3, -2, -13], [-2, 7, 0], [1, 5, -13])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2, -13], [-2, 7, 0], [1, 5, -13])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0017","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a06b512dba04999a5d3810805d77c72588dece39f16672871a0c29cba1d397f","family_id":"qf-str22-vectorspace-addition","seed":202610224017,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[4,-1,-14],\"right\":[-1,8,1]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[4, -1, -14], right=[-1, 8, 1]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([4, -1, -14], [-1, 8, 1], [3, 7, -13])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1, -14], [-1, 8, 1], [3, 7, -13])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0018","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a479b920d6eabe1843e32437dbb3a66adb50f2499a9fc0381ec5d764bcd2047b","family_id":"qf-str22-vectorspace-addition","seed":202610224018,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[5,0,-15],\"right\":[0,9,2]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[5, 0, -15], right=[0, 9, 2]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([5, 0, -15], [0, 9, 2], [5, 9, -13])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0, -15], [0, 9, 2], [5, 9, -13])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0019","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5abe5f10abbb56dcc5791350202c5c81631b16a04dc474f1cefaba2cec3bad86","family_id":"qf-str22-vectorspace-addition","seed":202610224019,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[6,1,-16],\"right\":[1,10,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[6, 1, -16], right=[1, 10, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([6, 1, -16], [1, 10, 0], [7, 11, -16])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1, -16], [1, 10, 0], [7, 11, -16])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0020","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05c93a6e41b70a27987cb9a51dd32c344536cd9967d80feadeba327470c2731f","family_id":"qf-str22-vectorspace-addition","seed":202610224020,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[7,2,-17],\"right\":[2,11,1]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[7, 2, -17], right=[2, 11, 1]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([7, 2, -17], [2, 11, 1], [9, 13, -16])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2, -17], [2, 11, 1], [9, 13, -16])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0021","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f68feffebfa8d849e16e03bf8783b2b96055f23d6edd2fa9228c4a1e99f4a83","family_id":"qf-str22-vectorspace-addition","seed":202610224021,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[8,3,-18],\"right\":[-2,12,2]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[8, 3, -18], right=[-2, 12, 2]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([8, 3, -18], [-2, 12, 2], [6, 15, -16])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 3, -18], [-2, 12, 2], [6, 15, -16])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0022","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6a80ef1cc004a86fe2e9371683937e6840a5d44b8ab7b4160b9e3e3989c9db0","family_id":"qf-str22-vectorspace-addition","seed":202610224022,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[9,-3,-19],\"right\":[-1,13,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[9, -3, -19], right=[-1, 13, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([9, -3, -19], [-1, 13, 0], [8, 10, -19])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -3, -19], [-1, 13, 0], [8, 10, -19])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0023","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"339afe286989a8f805b01c57625cc28f93c98592a5717f65aeebc55899930b6b","family_id":"qf-str22-vectorspace-addition","seed":202610224023,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[10,-2,-20],\"right\":[0,14,1]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[10, -2, -20], right=[0, 14, 1]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([10, -2, -20], [0, 14, 1], [10, 12, -19])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, -2, -20], [0, 14, 1], [10, 12, -19])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0024","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ee7f8a555e369a044a8d27a8d612182b1fbedfe7e7001e11bcfebb9ba546448","family_id":"qf-str22-vectorspace-addition","seed":202610224024,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[11,-1,-21],\"right\":[1,15,2]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[11, -1, -21], right=[1, 15, 2]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([11, -1, -21], [1, 15, 2], [12, 14, -19])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, -1, -21], [1, 15, 2], [12, 14, -19])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0025","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8db3c9d19c273dcd8eccde5151c3cdf96fd4d0ffb4d3ca53f1e9bb22716ac827","family_id":"qf-str22-vectorspace-addition","seed":202610224025,"semantic_template":"qf-str22-vectorspace-addition/{\"left\":[12,0,-22],\"right\":[2,16,0]}"},"fields":{"title":"Coordinate Addition in a Fixed Real Space","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[12, 0, -22], right=[2, 16, 0]. Give (left,right,the componentwise sum).","answer":{"kind":"expression","value":"([12, 0, -22], [2, 16, 0], [14, 16, -22])","explanation":"Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 0, -22], [2, 16, 0], [14, 16, -22])."},"skills":["Coordinate Addition in a Fixed Real Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0026","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3471151770c47058c470fc317bf9819a0366437629c7e12264bc4caffe2010d","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224026,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[-12,-2,3]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[-12, -2, 3]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-3, [-12, -2, 3], [36, 6, -9])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [-12, -2, 3], [36, 6, -9])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0027","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55b62811d3e61b7bf7811a9f9a177cbff3286b21d71a56c9adb9415f18ec5e38","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224027,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-2,\"vector\":[-11,-1,2]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[-11, -1, 2]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-2, [-11, -1, 2], [22, 2, -4])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [-11, -1, 2], [22, 2, -4])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0028","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4348c2a573ede755f0565299389f1a8840945cea8e6a9c2c71591e7f98d0bbf","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224028,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-1,\"vector\":[-10,0,1]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[-10, 0, 1]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-1, [-10, 0, 1], [10, 0, -1])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [-10, 0, 1], [10, 0, -1])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0029","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d69b18ec6a4a47abe33da2274c5f4145c8477b36af403d5cb2245786c9a696d","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224029,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":0,\"vector\":[-9,1,0]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[-9, 1, 0]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(0, [-9, 1, 0], [0, 0, 0])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [-9, 1, 0], [0, 0, 0])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0030","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"039658b7dd9cda741b3c2cc645bf8e7ae8b1f18a201d27b2bec3979651ebc3ce","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224030,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":1,\"vector\":[-8,2,-1]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[-8, 2, -1]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(1, [-8, 2, -1], [-8, 2, -1])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [-8, 2, -1], [-8, 2, -1])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0031","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fe5d4031d84c7b065330e61d2922eaa5aff4d698f20ab78be2360d1eef1e081","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224031,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":2,\"vector\":[-7,-2,-2]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[-7, -2, -2]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(2, [-7, -2, -2], [-14, -4, -4])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [-7, -2, -2], [-14, -4, -4])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0032","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e3eae8b87c9443667c3a35f111a54534e82063627c4ec24a18ee931256a4b76","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224032,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":3,\"vector\":[-6,-1,-3]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[-6, -1, -3]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(3, [-6, -1, -3], [-18, -3, -9])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [-6, -1, -3], [-18, -3, -9])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0033","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e7ef9f80ef2a9278a433f73886760a85ccdcfb48c64ce3b80106c59ae4579b2","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224033,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[-5,0,-4]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[-5, 0, -4]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-3, [-5, 0, -4], [15, 0, 12])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [-5, 0, -4], [15, 0, 12])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0034","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e78639d6be005735abaf26727ed6a55abaebae2a438a567897f7376e82a7fa22","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224034,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-2,\"vector\":[-4,1,-5]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[-4, 1, -5]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-2, [-4, 1, -5], [8, -2, 10])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [-4, 1, -5], [8, -2, 10])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0035","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f138309ac8fa8858cddbfdb7520aec0e7cfe40f9ac9c7e95542df940a51e2549","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224035,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-1,\"vector\":[-3,2,-6]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[-3, 2, -6]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-1, [-3, 2, -6], [3, -2, 6])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [-3, 2, -6], [3, -2, 6])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0036","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00ccec8122767462fb05c67720e1c8942e21d0ac3c634e5c92fe032ddce247b6","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224036,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":0,\"vector\":[-2,-2,-7]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[-2, -2, -7]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(0, [-2, -2, -7], [0, 0, 0])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [-2, -2, -7], [0, 0, 0])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0037","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58b8a7e3538739ded67cb89ed6f0dad30cbdc18f134287c6bf323b6a4d864e84","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224037,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":1,\"vector\":[-1,-1,-8]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[-1, -1, -8]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(1, [-1, -1, -8], [-1, -1, -8])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [-1, -1, -8], [-1, -1, -8])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0038","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62667903723500ada4603c4040746e7b4750be931cd3813f18785aec5b9a9c7f","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224038,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":2,\"vector\":[0,0,-9]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[0, 0, -9]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(2, [0, 0, -9], [0, 0, -18])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [0, 0, -9], [0, 0, -18])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0039","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de2cbbc0b20621d0e129b9b6d168977de1ef023e777ca99131be9d8e5042f34a","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224039,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":3,\"vector\":[1,1,-10]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[1, 1, -10]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(3, [1, 1, -10], [3, 3, -30])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [1, 1, -10], [3, 3, -30])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0040","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c760076de8d2b21cb5f2a1e554de2004655b562e33428accf617ac72abfa90d","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224040,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[2,2,-11]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[2, 2, -11]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-3, [2, 2, -11], [-6, -6, 33])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [2, 2, -11], [-6, -6, 33])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0041","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52f3813aaaebe0c56d0d280858c1315a484ee5aa28dcc0ae2625b5d9b5457c3d","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224041,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-2,\"vector\":[3,-2,-12]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[3, -2, -12]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-2, [3, -2, -12], [-6, 4, 24])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [3, -2, -12], [-6, 4, 24])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0042","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e30358f948fd630d714cb0b26ac3d942f4bcab0dca9e0b92deb7df91086b457","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224042,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-1,\"vector\":[4,-1,-13]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[4, -1, -13]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-1, [4, -1, -13], [-4, 1, 13])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [4, -1, -13], [-4, 1, 13])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0043","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c2ecedc915b4d6a7dbfe64017d577407c3e58b9ef388c82d639984391c7f83d","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224043,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":0,\"vector\":[5,0,-14]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[5, 0, -14]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(0, [5, 0, -14], [0, 0, 0])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [5, 0, -14], [0, 0, 0])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0044","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea5498abc9a15d3db9ddc79bc8abcb528226fcef36873fcabb1d3e2363cac751","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224044,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":1,\"vector\":[6,1,-15]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[6, 1, -15]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(1, [6, 1, -15], [6, 1, -15])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [6, 1, -15], [6, 1, -15])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0045","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb879bc5f755c417658742e5884740cc94560a79bc3437400f36a29b9af7a75f","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224045,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":2,\"vector\":[7,2,-16]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[7, 2, -16]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(2, [7, 2, -16], [14, 4, -32])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [7, 2, -16], [14, 4, -32])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0046","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71fc7caea78365446edefdd4ec82ee562b735cde83b35d99fb0214e567b7704d","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224046,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":3,\"vector\":[8,-2,-17]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[8, -2, -17]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(3, [8, -2, -17], [24, -6, -51])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [8, -2, -17], [24, -6, -51])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0047","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43a2f24e66617a100ed59cb12a2d20814bbb8c1b6807dbe55cbb9299108024a6","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224047,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-3,\"vector\":[9,-1,-18]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[9, -1, -18]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-3, [9, -1, -18], [-27, 3, 54])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [9, -1, -18], [-27, 3, 54])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0048","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88de12756780187c36bcc39d1757ef4da30a6a9df03a3417f41a18c9628d40c5","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224048,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-2,\"vector\":[10,0,-19]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[10, 0, -19]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-2, [10, 0, -19], [-20, 0, 38])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [10, 0, -19], [-20, 0, 38])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0049","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b36358a60005061a04e431cea0523188eb5b0cb32a06d061d046af18d4c3c09a","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224049,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":-1,\"vector\":[11,1,-20]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[11, 1, -20]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(-1, [11, 1, -20], [-11, -1, 20])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [11, 1, -20], [-11, -1, 20])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0050","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa4bef36857d5160f12890c417912f2fb19799b792dead1e228e0b6a5a4c1c0c","family_id":"qf-str22-vectorspace-scalar-multiplication","seed":202610224050,"semantic_template":"qf-str22-vectorspace-scalar-multiplication/{\"scalar\":0,\"vector\":[12,2,-21]}"},"fields":{"title":"A Declared Real-Scalar Instance on Integer Coordinates","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[12, 2, -21]. Give (scalar,vector,scaled vector).","answer":{"kind":"expression","value":"(0, [12, 2, -21], [0, 0, 0])","explanation":"Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [12, 2, -21], [0, 0, 0])."},"skills":["A Declared Real-Scalar Instance on Integer Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0051","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7da2061d32915af5a88d181b5a501c01c02b470ac4f03cc0c9f20ffa04bd524","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224051,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-12,-2,3]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-12, -2, 3]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-12, -2, 3], [12, 2, -3], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -2, 3], [12, 2, -3], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0052","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47160ce35b320f51a2ffa4ab07b6c479f20c440b7eacbd4a9a221fe3c1b394c7","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224052,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-11,-1,2]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-11, -1, 2]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-11, -1, 2], [11, 1, -2], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -1, 2], [11, 1, -2], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0053","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9a2e276fba804ab6ccd984f7985ea1647bae86123da564ca7873c0d3adc588e","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224053,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-10,0,1]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-10, 0, 1]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-10, 0, 1], [10, 0, -1], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, 0, 1], [10, 0, -1], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0054","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ba9439718aff6f24ac26b1084222347eb0b74bf0a09856c795bae176d7fe64d","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224054,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-9,1,0]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-9, 1, 0]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-9, 1, 0], [9, -1, 0], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 1, 0], [9, -1, 0], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0055","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62bb537fbbf7bac04f375ce2345c4052e8dc48f23b8e32decbd2f3ccfb2b7217","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224055,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-8,2,-1]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-8, 2, -1]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-8, 2, -1], [8, -2, 1], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 2, -1], [8, -2, 1], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0056","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e920b168c555c6eae4e3fddcab30a38b21e95b245c93ce79fb2aba37f61c82a2","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224056,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-7,-2,-2]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-7, -2, -2]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-7, -2, -2], [7, 2, 2], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, -2, -2], [7, 2, 2], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0057","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb0c117c8bf37a5cb31a9e745a09b2def0a4efed67279e6cf6872bb1a69835c8","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224057,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-6,-1,-3]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-6, -1, -3]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-6, -1, -3], [6, 1, 3], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, -1, -3], [6, 1, 3], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0058","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79452e81b469b6c1b3ed72b620f513e3de1f7380467b411153c04bff79149320","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224058,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-5,0,-4]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-5, 0, -4]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-5, 0, -4], [5, 0, 4], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, 0, -4], [5, 0, 4], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0059","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37f4d56fbfd056a93a867608f2d866732d2bdb55fb390c4715d91290cbc74f40","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224059,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-4,1,-5]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-4, 1, -5]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-4, 1, -5], [4, -1, 5], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, 1, -5], [4, -1, 5], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0060","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"759cb85380066a3ad6e99ff3d98db252bb0f4cc53fd7c885458ef178e7688eac","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224060,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-3,2,-6]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-3, 2, -6]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-3, 2, -6], [3, -2, 6], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, 2, -6], [3, -2, 6], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0061","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed82f122446695ec3d6075f9ca105e0a68043ef05a471a28bf13e36f7747453d","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224061,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-2,-2,-7]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-2, -2, -7]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-2, -2, -7], [2, 2, 7], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, -2, -7], [2, 2, 7], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0062","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c2bb2eb1134f8a680688b1fc8894433ebad6dd3c7b97b7eeb9d2aeb9d8bef97","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224062,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[-1,-1,-8]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-1, -1, -8]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([-1, -1, -8], [1, 1, 8], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, -1, -8], [1, 1, 8], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0063","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb8b5b029eaaa6ec8e50f9f857ed0a25099a2992162d5c3087964fb5653ad158","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224063,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[0,0,-9]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[0, 0, -9]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([0, 0, -9], [0, 0, 9], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 0, -9], [0, 0, 9], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0064","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"842e50b52358b56f15af5d08a65e8ad1989f1e4738014f7b1aa311254187fff7","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224064,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[1,1,-10]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[1, 1, -10]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([1, 1, -10], [-1, -1, 10], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 1, -10], [-1, -1, 10], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0065","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80716a6b15206719788f7bbb55f71b0ea82fd60e087e0902db0522998899baf0","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224065,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[2,2,-11]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[2, 2, -11]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([2, 2, -11], [-2, -2, 11], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 2, -11], [-2, -2, 11], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0066","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81411d6728c881ad2b89c98a3daae192c6d668f1b31d687027921bb42ada89ff","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224066,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[3,-2,-12]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[3, -2, -12]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([3, -2, -12], [-3, 2, 12], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2, -12], [-3, 2, 12], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0067","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11d81ccf7aad8640ec7eca47f21d477fc8ad70d72f554ed450441bca0e685ef3","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224067,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[4,-1,-13]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[4, -1, -13]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([4, -1, -13], [-4, 1, 13], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1, -13], [-4, 1, 13], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0068","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09a7a0763db9ad5f2efae25a288367f4f8f92ac9fe697f6a25166013237c995b","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224068,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[5,0,-14]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[5, 0, -14]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([5, 0, -14], [-5, 0, 14], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0, -14], [-5, 0, 14], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0069","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a694c447c474c40f7973afd76fde3c640d165ea3e72e34d325aab0b527acbe3","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224069,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[6,1,-15]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[6, 1, -15]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([6, 1, -15], [-6, -1, 15], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1, -15], [-6, -1, 15], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0070","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8eded8814c482e42a4b06a0e4afce9b3a6ebae93b0c4bf8dec6f82febb882f7","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224070,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[7,2,-16]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[7, 2, -16]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([7, 2, -16], [-7, -2, 16], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2, -16], [-7, -2, 16], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0071","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95e67aeb5383630d1d8ba9bcebbc77aa6d57c2a694c4c0bb6164820d541ad965","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224071,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[8,-2,-17]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[8, -2, -17]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([8, -2, -17], [-8, 2, 17], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, -2, -17], [-8, 2, 17], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0072","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f00e72a2f3e769f734572a2b7aa6eadda5d2bb8fcc217dfbee4c76f613b1662f","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224072,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[9,-1,-18]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[9, -1, -18]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([9, -1, -18], [-9, 1, 18], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -1, -18], [-9, 1, 18], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0073","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce651a0f62c70d022290cb26b53005b7baaee2f693986c262d262503d33bd309","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224073,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[10,0,-19]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[10, 0, -19]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([10, 0, -19], [-10, 0, 19], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 0, -19], [-10, 0, 19], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0074","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9db9c10be4db16d12eeef7b68e1e62020438eefdab8953fa43343d954fd6dabf","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224074,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[11,1,-20]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[11, 1, -20]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([11, 1, -20], [-11, -1, 20], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 1, -20], [-11, -1, 20], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0075","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8a1a1322e03a4d57373bac5e2b81fc76ad2eea624b5beb4c15dc084cfe5591e","family_id":"qf-str22-vectorspace-additive-inverse","seed":202610224075,"semantic_template":"qf-str22-vectorspace-additive-inverse/{\"vector\":[12,2,-21]}"},"fields":{"title":"The Coordinate Additive Inverse and Its Zero Sum","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[12, 2, -21]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).","answer":{"kind":"expression","value":"([12, 2, -21], [-12, -2, 21], [0, 0, 0], Yes)","explanation":"Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 2, -21], [-12, -2, 21], [0, 0, 0], Yes)."},"skills":["The Coordinate Additive Inverse and Its Zero Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0076","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"629a1c766bf816fc7791cb81787a55a52825e695fd85807aef0494d8845b94dc","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224076,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-3,\"b\":-2,\"vector\":[-12,-1,2]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=-2, vector=[-12, -1, 2]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-3, -2, [-12, -1, 2], [60, 5, -10], [60, 5, -10], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, -2, [-12, -1, 2], [60, 5, -10], [60, 5, -10], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0077","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f74510dee929899fbc9b986d75ed99dd927f73e920cca2b30791891413c4a4d","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224077,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-2,\"b\":-1,\"vector\":[-11,0,1]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=-1, vector=[-11, 0, 1]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-2, -1, [-11, 0, 1], [33, 0, -3], [33, 0, -3], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, -1, [-11, 0, 1], [33, 0, -3], [33, 0, -3], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0078","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5cffcd98b360681e4d246cc1d068643923e34c1fc2b0f2e96143dfb072b1dc7","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224078,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-1,\"b\":0,\"vector\":[-10,1,0]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=0, vector=[-10, 1, 0]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-1, 0, [-10, 1, 0], [10, -1, 0], [10, -1, 0], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 0, [-10, 1, 0], [10, -1, 0], [10, -1, 0], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0079","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"568463686de749c51d2bf2a3a92d1223224870db9884726153babd2eab1183a3","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224079,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":0,\"b\":1,\"vector\":[-9,2,-1]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=1, vector=[-9, 2, -1]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(0, 1, [-9, 2, -1], [-9, 2, -1], [-9, 2, -1], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 1, [-9, 2, -1], [-9, 2, -1], [-9, 2, -1], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0080","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cc0528885ae139cba2161c4d293cb38994fc29e60579bf7a4bc0fb59f5d3425","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224080,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":1,\"b\":2,\"vector\":[-8,-1,-2]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=2, vector=[-8, -1, -2]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(1, 2, [-8, -1, -2], [-24, -3, -6], [-24, -3, -6], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, 2, [-8, -1, -2], [-24, -3, -6], [-24, -3, -6], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0081","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5ec862268b75c717cf4c5cdab3e1ff52571da3096b393f02a477353f35acba9","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224081,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":2,\"b\":-2,\"vector\":[-7,0,-3]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=-2, vector=[-7, 0, -3]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(2, -2, [-7, 0, -3], [0, 0, 0], [0, 0, 0], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, -2, [-7, 0, -3], [0, 0, 0], [0, 0, 0], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0082","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2039ab423ddd4a03c2dae56a3649f29349ee3c5b757469f1bdec34b2651ce194","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224082,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":3,\"b\":-1,\"vector\":[-6,1,-4]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=-1, vector=[-6, 1, -4]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(3, -1, [-6, 1, -4], [-12, 2, -8], [-12, 2, -8], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, -1, [-6, 1, -4], [-12, 2, -8], [-12, 2, -8], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0083","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d03d68fb07fd2e062a87cfbdc8ad46d6c2a6680df23e544163b143233af5198e","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224083,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-3,\"b\":0,\"vector\":[-5,2,-5]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=0, vector=[-5, 2, -5]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-3, 0, [-5, 2, -5], [15, -6, 15], [15, -6, 15], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, 0, [-5, 2, -5], [15, -6, 15], [15, -6, 15], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0084","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d16138b114b911bc915dbe331f0c7530edf34666ec301bf64e1474590bc4ed6","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224084,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-2,\"b\":1,\"vector\":[-4,-1,-6]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=1, vector=[-4, -1, -6]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-2, 1, [-4, -1, -6], [4, 1, 6], [4, 1, 6], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, 1, [-4, -1, -6], [4, 1, 6], [4, 1, 6], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0085","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0acc91ed4356c8702d9b81c76627bd4ccc220337796974932d599743b43627fe","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224085,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-1,\"b\":2,\"vector\":[-3,0,-7]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=2, vector=[-3, 0, -7]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-1, 2, [-3, 0, -7], [-3, 0, -7], [-3, 0, -7], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 2, [-3, 0, -7], [-3, 0, -7], [-3, 0, -7], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0086","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45b67641812d12005de6b43a75f8863be0d25ff0225f7eebf223cc21b50e0d7c","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224086,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":0,\"b\":-2,\"vector\":[-2,1,-8]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=-2, vector=[-2, 1, -8]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(0, -2, [-2, 1, -8], [4, -2, 16], [4, -2, 16], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, -2, [-2, 1, -8], [4, -2, 16], [4, -2, 16], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0087","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5226a16237ac7d8684b798b8ad6ca66aa68342cecfd7c600b6afde15541eba28","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224087,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":1,\"b\":-1,\"vector\":[-1,2,-9]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=-1, vector=[-1, 2, -9]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(1, -1, [-1, 2, -9], [0, 0, 0], [0, 0, 0], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, -1, [-1, 2, -9], [0, 0, 0], [0, 0, 0], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0088","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b87e6735976b9d19a0f09c7c2d942fc38b5ea22841bfb568b5ad7fa69a33daa","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224088,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":2,\"b\":0,\"vector\":[0,-1,-10]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=0, vector=[0, -1, -10]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(2, 0, [0, -1, -10], [0, -2, -20], [0, -2, -20], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, 0, [0, -1, -10], [0, -2, -20], [0, -2, -20], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0089","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf809cc8fd62f1db469f37f7149c3ad3ffb5c1a3ce1b87be8343e977964c599d","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224089,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":3,\"b\":1,\"vector\":[1,0,-11]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=1, vector=[1, 0, -11]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(3, 1, [1, 0, -11], [4, 0, -44], [4, 0, -44], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, 1, [1, 0, -11], [4, 0, -44], [4, 0, -44], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0090","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc1d1f42550f93058c2779f3a103859e008ebff86abf43fd15ff9d274d669620","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224090,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-3,\"b\":2,\"vector\":[2,1,-12]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=2, vector=[2, 1, -12]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-3, 2, [2, 1, -12], [-2, -1, 12], [-2, -1, 12], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, 2, [2, 1, -12], [-2, -1, 12], [-2, -1, 12], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0091","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ffb199250b122185770b9be053cffb883e49d80d2d1cd40f4fc86a3deb5413a","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224091,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-2,\"b\":-2,\"vector\":[3,2,-13]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=-2, vector=[3, 2, -13]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-2, -2, [3, 2, -13], [-12, -8, 52], [-12, -8, 52], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, -2, [3, 2, -13], [-12, -8, 52], [-12, -8, 52], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0092","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77aba5d7e18e0aff49b3d6cdffb8426b5e1ced8722820b9f01cdeb99cd2f2d61","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224092,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-1,\"b\":-1,\"vector\":[4,-1,-14]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=-1, vector=[4, -1, -14]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-1, -1, [4, -1, -14], [-8, 2, 28], [-8, 2, 28], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, -1, [4, -1, -14], [-8, 2, 28], [-8, 2, 28], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0093","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"868808197ea45e834b104eae17e2692dc8f42a60aa3b4d88823accdec608c2ac","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224093,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":0,\"b\":0,\"vector\":[5,0,-15]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=0, vector=[5, 0, -15]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0094","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58d5fcb9c4bb57aceb1d5d3c94941c4c8dae0f0315c279a35a74f0f39dc001e7","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224094,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":1,\"b\":1,\"vector\":[6,1,-16]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=1, vector=[6, 1, -16]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(1, 1, [6, 1, -16], [12, 2, -32], [12, 2, -32], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, 1, [6, 1, -16], [12, 2, -32], [12, 2, -32], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0095","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b49c1f671f87fb887b3a99baba04f1e4bc9e227a4f2edd8545bfcada3543882","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224095,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":2,\"b\":2,\"vector\":[7,2,-17]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=2, vector=[7, 2, -17]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(2, 2, [7, 2, -17], [28, 8, -68], [28, 8, -68], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, 2, [7, 2, -17], [28, 8, -68], [28, 8, -68], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0096","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16c0cdf52b9f5ee6e839efe162370c433917a6c652804d4fa77e0424e2999b96","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224096,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":3,\"b\":-2,\"vector\":[8,-1,-18]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=-2, vector=[8, -1, -18]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(3, -2, [8, -1, -18], [8, -1, -18], [8, -1, -18], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, -2, [8, -1, -18], [8, -1, -18], [8, -1, -18], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0097","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6174fa7c241fe2da9968de07c5a2e90b67ad55057b3c2b98b6544917589c2ab2","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224097,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-3,\"b\":-1,\"vector\":[9,0,-19]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=-1, vector=[9, 0, -19]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-3, -1, [9, 0, -19], [-36, 0, 76], [-36, 0, 76], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, -1, [9, 0, -19], [-36, 0, 76], [-36, 0, 76], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0098","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21422e034848d61dd40dd668009f984c97106d133b52600bf32be41a643dfe92","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224098,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-2,\"b\":0,\"vector\":[10,1,-20]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=0, vector=[10, 1, -20]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-2, 0, [10, 1, -20], [-20, -2, 40], [-20, -2, 40], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, 0, [10, 1, -20], [-20, -2, 40], [-20, -2, 40], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0099","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a980b691b3e18ea27f265e2cad883e272d991206a265602c0b74c86242db5a4","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224099,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":-1,\"b\":1,\"vector\":[11,2,-21]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=1, vector=[11, 2, -21]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(-1, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r04-real-vector-space-0100","batch_id":"batch-20261013-advance-20261022-r04-real-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"392f5384687bd5c8913bc1ccba4910cb4f89b53b34af76644a803c947f5734fb","family_id":"qf-str22-vectorspace-scalar-distributivity","seed":202610224100,"semantic_template":"qf-str22-vectorspace-scalar-distributivity/{\"a\":0,\"b\":2,\"vector\":[12,-1,-22]}"},"fields":{"title":"One Exact Instance of Distributivity over Scalar Addition","stem":"Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=2, vector=[12, -1, -22]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).","answer":{"kind":"expression","value":"(0, 2, [12, -1, -22], [24, -2, -44], [24, -2, -44], Yes)","explanation":"Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 2, [12, -1, -22], [24, -2, -44], [24, -2, -44], Yes)."},"skills":["One Exact Instance of Distributivity over Scalar Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}