{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0001","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d4c03c794b8f451119cfe3be4bf8a1b41e6c9ea92e4f15ef8ad590c7745dad5","family_id":"qf-mod36-directsum-internal-audit","seed":202611051001,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,1]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 1]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0002","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2b74ccd37c55ccb3cf412c3b70ee192734689cdce37ceb3ae9f8275edb30211","family_id":"qf-mod36-directsum-internal-audit","seed":202611051002,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,2]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 2]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0003","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ecb7223c58a06e6ead3fd09109e65694f6bdf37c3b596c8995527b25cd058a4","family_id":"qf-mod36-directsum-internal-audit","seed":202611051003,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,3]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 3]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0004","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41b381c19d25f2fed7e6cf3899b1f79c41862e4209fc9e9377139d2f1b20093f","family_id":"qf-mod36-directsum-internal-audit","seed":202611051004,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,4]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 4]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0005","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e76fbd49e581d7f4385bd78320af42a684630bb98c3a3a90b6ccd4412f32e6af","family_id":"qf-mod36-directsum-internal-audit","seed":202611051005,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,5]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 5]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0006","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f97c839a5a032626e3e2243c59b7a16e7a8486d7ea6c33262382293c80b6ee9e","family_id":"qf-mod36-directsum-internal-audit","seed":202611051006,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,6]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 6]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0007","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e3ef402bc48e99680c2cc8c117febeeb0c5479b0b46437adda5beb79d8bfbf4","family_id":"qf-mod36-directsum-internal-audit","seed":202611051007,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,7]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 7]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0008","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24e528e2c11c7a372cbeb041998c96f8c7eeae970f8db636103527f55fcf02c0","family_id":"qf-mod36-directsum-internal-audit","seed":202611051008,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,8]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 8]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0009","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f65c70b49337f4a384a56d983c39267e55d5980161170b0bdcf1506bdc37d60","family_id":"qf-mod36-directsum-internal-audit","seed":202611051009,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,9]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 9]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0010","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa27b7b05af3135f7dbb122161c5f6f7f827922eae8d74bde804b012e90825de","family_id":"qf-mod36-directsum-internal-audit","seed":202611051010,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,10]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 10]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0011","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98a4e6f1853a0b096f0117aec0aac4a1221608ddde96dcb0e0a2079e97ba9a71","family_id":"qf-mod36-directsum-internal-audit","seed":202611051011,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,11]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 11]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0012","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c95c4f8a1732a32ce28750d7e04f4e104ca265a9084e93189d0ec4f313ef745","family_id":"qf-mod36-directsum-internal-audit","seed":202611051012,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,12]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 12]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0013","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b566e48c7f117d2305650bcbfb516457aeb8421e35aa08ba2e5eeccc4c16c1a1","family_id":"qf-mod36-directsum-internal-audit","seed":202611051013,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,13]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 13]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0014","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e26eb26cffeb0fd16ae5ab531d0864cd0d8e2406a4a7bab051cc968206083a6f","family_id":"qf-mod36-directsum-internal-audit","seed":202611051014,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,14]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 14]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0015","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d076ab86dc229bcf9fba8b13a9caa1be9d2889b68ec6372e19bb7094f2bed4d","family_id":"qf-mod36-directsum-internal-audit","seed":202611051015,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,15]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 15]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0016","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2837cba6027040c45412fb325ec93a779d7b7ceee2d18fd546c857e9b853c613","family_id":"qf-mod36-directsum-internal-audit","seed":202611051016,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,16]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 16]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0017","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fafebce65796bb8647296c0c621ee285d5dffc8d7ebb05faf1a3506681de74fe","family_id":"qf-mod36-directsum-internal-audit","seed":202611051017,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,17]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 17]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0018","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74e3884374a65b988f3e450a82ab169c4246577a9082bfd93e4065a6ededd3c8","family_id":"qf-mod36-directsum-internal-audit","seed":202611051018,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,18]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 18]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0019","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c66a1198a8ad4b83453da563233754c4645ecee237650c98dbb288d867bb3e50","family_id":"qf-mod36-directsum-internal-audit","seed":202611051019,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,19]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 19]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0020","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf580c916c272df32e631c6f61dcaeb16bc86ad03141f2f7694823ba81b03a13","family_id":"qf-mod36-directsum-internal-audit","seed":202611051020,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,20]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 20]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0021","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17d2fae28671f214d9a3450547533dbb99ea395b1da8cc6311556287c389d936","family_id":"qf-mod36-directsum-internal-audit","seed":202611051021,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,21]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 21]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0022","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6021b64873dc351e7534a0388127fa069da25c2c824fecdbcdd85abeb0fb0828","family_id":"qf-mod36-directsum-internal-audit","seed":202611051022,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,22]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 22]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0023","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bb8bfa77008572c89399b9d5416c9ae1e11bb3549b749295457cbdb48ea9cca","family_id":"qf-mod36-directsum-internal-audit","seed":202611051023,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,23]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 23]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0024","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb411379276b159e67e91be01cd193ecf4c3d02cc96aef5a8e3ef6b234d7d10","family_id":"qf-mod36-directsum-internal-audit","seed":202611051024,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,24]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 24]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0025","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d9b344f5e02ebda57e2e817da8e6a2f5bcbdb79105acbf06fcafb5ff80772ff","family_id":"qf-mod36-directsum-internal-audit","seed":202611051025,"semantic_template":"qf-mod36-directsum-internal-audit/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,25]]}"},"fields":{"title":"Uniqueness in a Sum Does Not Yet Give Ambient Coverage","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 25]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Give (dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum).","answer":{"kind":"expression","value":"(1, 1, 2, 0, Yes, No, No)","explanation":"Compute dimensions of U, W and U+W, then dim(U∩W)=dim(U)+dim(W)−dim(U+W). Zero intersection gives unique decomposition inside the sum. Only uniqueness together with full ambient coverage gives V=U⊕W. The exact result, retaining the requested order, is (1, 1, 2, 0, Yes, No, No)."},"skills":["Uniqueness in a Sum Does Not Yet Give Ambient Coverage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0026","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b67a47caeece1d995f8507d7044f703afc218eff50b930e115fd7cb8d2b054e5","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051026,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,1]],\"coefficients_u\":[1],\"coefficients_w\":[2],\"vector\":[1,2,2]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 1]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[1, 2, 2], coefficients_u=[1], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([1, 0, 0], [0, 2, 2], [1, 2, 2], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([1, 0, 0], [0, 2, 2], [1, 2, 2], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0027","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"036edc88d4225b27d1d9b95ff95a8760c4902ddf1d897e5f4efa41bf34de63a4","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051027,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,2]],\"coefficients_u\":[2],\"coefficients_w\":[2],\"vector\":[2,2,4]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 2]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[2, 2, 4], coefficients_u=[2], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([2, 0, 0], [0, 2, 4], [2, 2, 4], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([2, 0, 0], [0, 2, 4], [2, 2, 4], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0028","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a72c3693ef48a5e763cd12bcdec57a43df63943b4ba6fe99e3e266ab4b41867","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051028,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,3]],\"coefficients_u\":[3],\"coefficients_w\":[2],\"vector\":[3,2,6]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 3]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[3, 2, 6], coefficients_u=[3], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([3, 0, 0], [0, 2, 6], [3, 2, 6], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([3, 0, 0], [0, 2, 6], [3, 2, 6], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0029","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ce64c5e54dbd24e1e22d39d0e99271efedeb9aef5f5023aa50fc63bef84510d","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051029,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,4]],\"coefficients_u\":[4],\"coefficients_w\":[2],\"vector\":[4,2,8]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 4]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[4, 2, 8], coefficients_u=[4], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([4, 0, 0], [0, 2, 8], [4, 2, 8], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([4, 0, 0], [0, 2, 8], [4, 2, 8], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0030","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdcda8c69782573297c8e886a50c8541b46abcadb470c5b6052d7decc076a902","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051030,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,5]],\"coefficients_u\":[5],\"coefficients_w\":[2],\"vector\":[5,2,10]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 5]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[5, 2, 10], coefficients_u=[5], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([5, 0, 0], [0, 2, 10], [5, 2, 10], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([5, 0, 0], [0, 2, 10], [5, 2, 10], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0031","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08827d4bf92e3884b17531aafadbad80c3747d69496dec1850e8cbe4a985e0c2","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051031,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,6]],\"coefficients_u\":[6],\"coefficients_w\":[2],\"vector\":[6,2,12]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 6]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[6, 2, 12], coefficients_u=[6], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([6, 0, 0], [0, 2, 12], [6, 2, 12], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([6, 0, 0], [0, 2, 12], [6, 2, 12], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0032","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11a8230b49da2bf1158259695797e907a9833d3d86566f6133e5247ff352f011","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051032,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,7]],\"coefficients_u\":[7],\"coefficients_w\":[2],\"vector\":[7,2,14]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 7]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[7, 2, 14], coefficients_u=[7], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([7, 0, 0], [0, 2, 14], [7, 2, 14], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([7, 0, 0], [0, 2, 14], [7, 2, 14], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0033","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc65b6b5849ae80cfc12fa4738f47762ed1c3b60a6eb67385da6ab18491d2219","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051033,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,8]],\"coefficients_u\":[8],\"coefficients_w\":[2],\"vector\":[8,2,16]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 8]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[8, 2, 16], coefficients_u=[8], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([8, 0, 0], [0, 2, 16], [8, 2, 16], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([8, 0, 0], [0, 2, 16], [8, 2, 16], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0034","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32e87d154ca710525f972d925e3378d4f2e46475988b6034f009c3be4b3e47b4","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051034,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,9]],\"coefficients_u\":[9],\"coefficients_w\":[2],\"vector\":[9,2,18]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 9]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[9, 2, 18], coefficients_u=[9], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([9, 0, 0], [0, 2, 18], [9, 2, 18], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([9, 0, 0], [0, 2, 18], [9, 2, 18], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0035","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fed3797f3a3f5338dfcb4a60ee6579d789388c4829e487e0daa00f0d0438a633","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051035,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,10]],\"coefficients_u\":[10],\"coefficients_w\":[2],\"vector\":[10,2,20]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 10]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[10, 2, 20], coefficients_u=[10], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([10, 0, 0], [0, 2, 20], [10, 2, 20], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([10, 0, 0], [0, 2, 20], [10, 2, 20], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0036","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56c03bb8bfd8ce52081f73084167f826b1ab6cbeac1717a92cc60d87da728a8b","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051036,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,11]],\"coefficients_u\":[11],\"coefficients_w\":[2],\"vector\":[11,2,22]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 11]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[11, 2, 22], coefficients_u=[11], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([11, 0, 0], [0, 2, 22], [11, 2, 22], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([11, 0, 0], [0, 2, 22], [11, 2, 22], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0037","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e380e083cc6b8b3735cf0b702987a1ad41090428de30710986086eef0bdabee","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051037,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,12]],\"coefficients_u\":[12],\"coefficients_w\":[2],\"vector\":[12,2,24]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 12]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[12, 2, 24], coefficients_u=[12], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([12, 0, 0], [0, 2, 24], [12, 2, 24], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([12, 0, 0], [0, 2, 24], [12, 2, 24], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0038","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8857a12b4b34682184ab2c0c62fe069f55c96486d08e7b71973144537d840d69","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051038,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,13]],\"coefficients_u\":[13],\"coefficients_w\":[2],\"vector\":[13,2,26]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 13]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[13, 2, 26], coefficients_u=[13], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([13, 0, 0], [0, 2, 26], [13, 2, 26], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([13, 0, 0], [0, 2, 26], [13, 2, 26], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0039","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ca39e4d424759449e31d58a246ae7b6c9a6ddff7a0bcbd38ab3a4339be62534","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051039,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,14]],\"coefficients_u\":[14],\"coefficients_w\":[2],\"vector\":[14,2,28]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 14]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[14, 2, 28], coefficients_u=[14], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([14, 0, 0], [0, 2, 28], [14, 2, 28], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([14, 0, 0], [0, 2, 28], [14, 2, 28], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0040","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7951ec761deb5f5bcc6c4b989d4075cabcf38dacaacd266d79781822c0ffa385","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051040,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,15]],\"coefficients_u\":[15],\"coefficients_w\":[2],\"vector\":[15,2,30]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 15]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[15, 2, 30], coefficients_u=[15], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([15, 0, 0], [0, 2, 30], [15, 2, 30], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([15, 0, 0], [0, 2, 30], [15, 2, 30], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0041","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e15deddf281002fc522fb48fe3c3d06da5a60bb547defb08604c593647ab3b5c","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051041,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,16]],\"coefficients_u\":[16],\"coefficients_w\":[2],\"vector\":[16,2,32]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 16]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[16, 2, 32], coefficients_u=[16], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([16, 0, 0], [0, 2, 32], [16, 2, 32], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([16, 0, 0], [0, 2, 32], [16, 2, 32], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0042","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7546dd436b5b96657e570e4b6c7fd7ea69082f1a2ae53b3ca0b0fc36902cb8a8","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051042,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,17]],\"coefficients_u\":[17],\"coefficients_w\":[2],\"vector\":[17,2,34]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 17]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[17, 2, 34], coefficients_u=[17], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([17, 0, 0], [0, 2, 34], [17, 2, 34], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([17, 0, 0], [0, 2, 34], [17, 2, 34], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0043","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5984bfcea266d35f3ac9fa825905d282b58e03512501e0dd8868c205d312754a","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051043,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,18]],\"coefficients_u\":[18],\"coefficients_w\":[2],\"vector\":[18,2,36]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 18]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[18, 2, 36], coefficients_u=[18], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([18, 0, 0], [0, 2, 36], [18, 2, 36], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([18, 0, 0], [0, 2, 36], [18, 2, 36], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0044","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1537b1cefa3244daafaea88cae6096b6be8f33d08858f92edb6dd3a6ae9b71fe","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051044,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,19]],\"coefficients_u\":[19],\"coefficients_w\":[2],\"vector\":[19,2,38]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 19]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[19, 2, 38], coefficients_u=[19], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([19, 0, 0], [0, 2, 38], [19, 2, 38], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([19, 0, 0], [0, 2, 38], [19, 2, 38], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0045","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18c288158d1336a99be3ffbdb82cd79a2a57ce812c7949282fbaac2b71a1d702","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051045,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,20]],\"coefficients_u\":[20],\"coefficients_w\":[2],\"vector\":[20,2,40]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 20]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[20, 2, 40], coefficients_u=[20], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([20, 0, 0], [0, 2, 40], [20, 2, 40], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([20, 0, 0], [0, 2, 40], [20, 2, 40], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0046","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffdf7623b35a8cda588bad89fbce0d66382e7d58ef735be49098ee1284c6f284","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051046,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,21]],\"coefficients_u\":[21],\"coefficients_w\":[2],\"vector\":[21,2,42]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 21]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[21, 2, 42], coefficients_u=[21], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([21, 0, 0], [0, 2, 42], [21, 2, 42], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([21, 0, 0], [0, 2, 42], [21, 2, 42], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0047","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ed561d017ff1071e5667609352e564391d09e63fd79a217e457fbaf1b903b22","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051047,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,22]],\"coefficients_u\":[22],\"coefficients_w\":[2],\"vector\":[22,2,44]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 22]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[22, 2, 44], coefficients_u=[22], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([22, 0, 0], [0, 2, 44], [22, 2, 44], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([22, 0, 0], [0, 2, 44], [22, 2, 44], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0048","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7850c270a99d7826ca3c18dec303ef343ca90dc1a7d97cb21b07d36fe8c21ed","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051048,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,23]],\"coefficients_u\":[23],\"coefficients_w\":[2],\"vector\":[23,2,46]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 23]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[23, 2, 46], coefficients_u=[23], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([23, 0, 0], [0, 2, 46], [23, 2, 46], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([23, 0, 0], [0, 2, 46], [23, 2, 46], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0049","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97407ed5167120113e37eead4a5c585cf77bf7a4b63bbd99a465a41f7fd3a39b","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051049,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,24]],\"coefficients_u\":[24],\"coefficients_w\":[2],\"vector\":[24,2,48]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 24]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[24, 2, 48], coefficients_u=[24], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([24, 0, 0], [0, 2, 48], [24, 2, 48], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([24, 0, 0], [0, 2, 48], [24, 2, 48], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0050","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c1ab799ff2a9a30c988f39ebe3d2330728d16b5aacef5475cb593f34c1397d1","family_id":"qf-mod36-directsum-ambient-decomposition","seed":202611051050,"semantic_template":"qf-mod36-directsum-ambient-decomposition/{\"ambient_dimension\":3,\"basis_u\":[[1,0,0]],\"basis_w\":[[0,1,25]],\"coefficients_u\":[25],\"coefficients_w\":[2],\"vector\":[25,2,50]}"},"fields":{"title":"Check This Decomposition Separately from a Whole-Space Direct Sum","stem":"Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 25]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[25, 2, 50], coefficients_u=[25], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"([25, 0, 0], [0, 2, 50], [25, 2, 50], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)","explanation":"Form both linear combinations in supplied basis order, add them and compare with the target. Retain the independent sum-uniqueness and ambient-coverage flags; successful reconstruction of this target does not prove coverage of every ambient vector. The exact result, retaining the requested order, is ([25, 0, 0], [0, 2, 50], [25, 2, 50], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)."},"skills":["Check This Decomposition Separately from a Whole-Space Direct 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-20261014-advance-20261105-r01-direct-sum-0051","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cfc2600cfa70dd7f86b1c6bb1a8471d0bf8458042b3c7b35f75e71b7dcd384d","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051051,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,1]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 1]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0052","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3be2b2928fdd3cb80b07ab4219b680a2635a188710e66035f0f793a87467c722","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051052,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,2]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 2]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0053","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d55e0a26336613eda5704e00afd0d2b448cecf0607dbc7b29e6de20958b19037","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051053,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,3]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 3]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0054","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1664f7daa3a40e0763c8dcdc9effd45bf21f23bc8b8ab83734a69e5f958bf57","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051054,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,4]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 4]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0055","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fc9eb72d140dcd7242a11a9463af4f8107ac77ae87471cc97600494f1e41a35","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051055,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,5]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 5]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0056","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a967b2d6bdc9050c8134e611210eccd8d6a09fa9ee4a18bd3b1e21d4c6f51e0c","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051056,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,6]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 6]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0057","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6781402e7a0c7ebc9f68a366d6503c5950ff1b8e23d519be0ac20c286e939186","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051057,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,7]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 7]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0058","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d20adc0c8952a4aa1509b62979799d5dec9882d0aa0e6471d6761fa4c01d181","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051058,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,8]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 8]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0059","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2bc356230f4aca9c7406b6931d9c91d8d5ab72aee34513e250a082437929ff3","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051059,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,9]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 9]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0060","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4aa50fd02fc3e1901f231a8e6dc308407fd5ddfabcaf4f3e8f1dc92ec49be02","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051060,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,10]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 10]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0061","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49370d480e0a365cc625a2d9773ae62f25746bb6b02f14807cbfba7390589c03","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051061,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,11]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 11]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0062","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5177bf1b9c4111b8e540c8c1c2b21e74644ec70084f893a06e7b474feffb8d98","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051062,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,12]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 12]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0063","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21f1f1d515ca2de067f23455c8fe99cdb84fe5aa8c524a1766cc227dc987ebb1","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051063,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,13]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 13]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0064","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ed3ece3ebb3f669bbc423565dbf53a0c0e95d3d5f5fed84b5a4de8d906a1006","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051064,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,14]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 14]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0065","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0344d26bbbc96e12865f6c0e70b992d970e1f89674514b99382b2d9e2c54dd89","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051065,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,15]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 15]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0066","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"606157c275192f94bf5ead541be87819dd11e711e9d85be96521ccd790e55ed3","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051066,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,16]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 16]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0067","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a9f517dbc39f3050d04722559175a1624c31b2ebe173e001cd24696e177e1d0","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051067,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,17]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 17]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0068","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04455f5ed93afe1e0999915b05f513a5cd88a8a481278f55c3b2ef5974d92380","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051068,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,18]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 18]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0069","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6e10fbdd73cdbd1cb3aa5b72733387a6af520656217972c0a6449cdf22cc85f","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051069,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,19]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 19]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0070","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe83b485a32049098773215025ef78fc0db9752ef51afcc2e04730142d2f8ddb","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051070,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,20]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 20]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0071","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35b4da90dda40bf43e5be4604f76fe14c1a5bfe1eb1ecb67f5bc720bbe9e6a19","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051071,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,21]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 21]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0072","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae83b939652da2bc9c5acf72751f7e94040b13c6e2ea535a7ac71b20f2b47796","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051072,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,22]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 22]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0073","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"570d585f7eab6502dff55749cb994eeebaf7019666076055d97c144cbb91b64b","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051073,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,23]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 23]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0074","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b7e1a5dc310a45820bb17163e0401b7b3f82368615d85bbfb9f2dfa8662938e","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051074,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,24]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 24]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0075","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91b17cec5c4a293b7c174680f4b9c4dd2740f1391f3c43a3366791b1711c397d","family_id":"qf-mod36-directsum-multi-subspace-global","seed":202611051075,"semantic_template":"qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,25]]]}"},"fields":{"title":"Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum","stem":"Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 25]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).","answer":{"kind":"expression","value":"(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)","explanation":"Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)."},"skills":["Pairwise Zero Intersections Do Not Prove a Three-Way Direct 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-20261014-advance-20261105-r01-direct-sum-0076","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d24f280c162aab67eb7ef41e91935ccd4240c61d438caca42a36defe1059a6b0","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051076,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[0,1],\"other_left\":[3,1],\"other_right\":[4],\"right\":[2],\"scalar\":1}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[0, 1], right=[2], another pair other_left=[3, 1], other_right=[4], and scalar=1. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[0, 1], [2]], [[3, 2], [6]], [[0, 1], [2]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[0, 1], [2]], [[3, 2], [6]], [[0, 1], [2]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0077","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2db0c09ab6cfd8f7c87eb904b179538f0d1779b7ea869a6a6c159a22fa3f2dcb","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051077,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[1,2],\"other_left\":[4,3],\"other_right\":[5],\"right\":[3],\"scalar\":2}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[1, 2], right=[3], another pair other_left=[4, 3], other_right=[5], and scalar=2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0078","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f81b7690021c23c700f26cdfd2e9f3bde42944e250849604d8821c2b7816f1d","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051078,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[2,3],\"other_left\":[5,5],\"other_right\":[6],\"right\":[4],\"scalar\":3}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[2, 3], right=[4], another pair other_left=[5, 5], other_right=[6], and scalar=3. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[2, 3], [4]], [[7, 8], [10]], [[6, 9], [12]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[2, 3], [4]], [[7, 8], [10]], [[6, 9], [12]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0079","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d3696975964864fd760859de211e4aeacb3e804d9dd9ab0a3e1f155ab5e4577","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051079,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[3,4],\"other_left\":[6,7],\"other_right\":[7],\"right\":[5],\"scalar\":4}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[3, 4], right=[5], another pair other_left=[6, 7], other_right=[7], and scalar=4. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[3, 4], [5]], [[9, 11], [12]], [[12, 16], [20]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[3, 4], [5]], [[9, 11], [12]], [[12, 16], [20]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0080","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c0abd134552b01e4ea70b436996740c383963eb767583dcff20ff4dc13a8c78","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051080,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[4,5],\"other_left\":[7,9],\"other_right\":[8],\"right\":[6],\"scalar\":5}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[4, 5], right=[6], another pair other_left=[7, 9], other_right=[8], and scalar=5. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[4, 5], [6]], [[11, 14], [14]], [[20, 25], [30]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[4, 5], [6]], [[11, 14], [14]], [[20, 25], [30]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0081","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aa8ed6f83d08a096ab8764a615632d4fce7d860ddd09d56f1777ef1342b71ea","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051081,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[5,6],\"other_left\":[8,11],\"other_right\":[9],\"right\":[7],\"scalar\":6}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[5, 6], right=[7], another pair other_left=[8, 11], other_right=[9], and scalar=6. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[5, 6], [7]], [[13, 17], [16]], [[30, 36], [42]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[5, 6], [7]], [[13, 17], [16]], [[30, 36], [42]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0082","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf03055edffe7e935044319859116fed27aac6c3af54ac123df031eac5eb0c00","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051082,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[6,7],\"other_left\":[9,13],\"other_right\":[10],\"right\":[8],\"scalar\":7}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[6, 7], right=[8], another pair other_left=[9, 13], other_right=[10], and scalar=7. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[6, 7], [8]], [[15, 20], [18]], [[42, 49], [56]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[6, 7], [8]], [[15, 20], [18]], [[42, 49], [56]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0083","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c52c5cd30f45ed2ce65b6883bd3b7e8da70106099eb141b3707a046e12bfe2c","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051083,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[7,8],\"other_left\":[10,15],\"other_right\":[11],\"right\":[9],\"scalar\":8}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[7, 8], right=[9], another pair other_left=[10, 15], other_right=[11], and scalar=8. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[7, 8], [9]], [[17, 23], [20]], [[56, 64], [72]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[7, 8], [9]], [[17, 23], [20]], [[56, 64], [72]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0084","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5f908906008c1f36da6030fd0a2072acfe3c8ce33fed1ab441b80a4b19a48ac","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051084,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[8,9],\"other_left\":[11,17],\"other_right\":[12],\"right\":[10],\"scalar\":9}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[8, 9], right=[10], another pair other_left=[11, 17], other_right=[12], and scalar=9. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[8, 9], [10]], [[19, 26], [22]], [[72, 81], [90]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[8, 9], [10]], [[19, 26], [22]], [[72, 81], [90]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0085","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a07de462e66af95ab548811ad7bc1641fc8e9734216da095cd19d734ce38ac46","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051085,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[9,10],\"other_left\":[12,19],\"other_right\":[13],\"right\":[11],\"scalar\":10}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[9, 10], right=[11], another pair other_left=[12, 19], other_right=[13], and scalar=10. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[9, 10], [11]], [[21, 29], [24]], [[90, 100], [110]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[9, 10], [11]], [[21, 29], [24]], [[90, 100], [110]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0086","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e78cf2eb913d5bb99ba56bd830a92eb4d4c864108c2b0d22be0021789429a78a","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051086,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[10,11],\"other_left\":[13,21],\"other_right\":[14],\"right\":[12],\"scalar\":11}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[10, 11], right=[12], another pair other_left=[13, 21], other_right=[14], and scalar=11. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[10, 11], [12]], [[23, 32], [26]], [[110, 121], [132]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[10, 11], [12]], [[23, 32], [26]], [[110, 121], [132]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0087","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89d1cc808e36eefff120da834126c07f570b4ab1724e6735ca19282a23e8fa3d","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051087,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[11,12],\"other_left\":[14,23],\"other_right\":[15],\"right\":[13],\"scalar\":12}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[11, 12], right=[13], another pair other_left=[14, 23], other_right=[15], and scalar=12. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[11, 12], [13]], [[25, 35], [28]], [[132, 144], [156]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[11, 12], [13]], [[25, 35], [28]], [[132, 144], [156]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0088","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6085cbd3c8b7d28f37b99a35e1c61181885334e3176f28d622a90f99b922bfc","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051088,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[12,13],\"other_left\":[15,25],\"other_right\":[16],\"right\":[14],\"scalar\":13}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[12, 13], right=[14], another pair other_left=[15, 25], other_right=[16], and scalar=13. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[12, 13], [14]], [[27, 38], [30]], [[156, 169], [182]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[12, 13], [14]], [[27, 38], [30]], [[156, 169], [182]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0089","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e27973015cb4bfc730992d11288df04e3b9a40deb1ff475e81f8bed83436cb01","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051089,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[13,14],\"other_left\":[16,27],\"other_right\":[17],\"right\":[15],\"scalar\":14}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[13, 14], right=[15], another pair other_left=[16, 27], other_right=[17], and scalar=14. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[13, 14], [15]], [[29, 41], [32]], [[182, 196], [210]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[13, 14], [15]], [[29, 41], [32]], [[182, 196], [210]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0090","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16a54feeded2de04cc50ccfb4236ca91fa17bdb420951b0fde5d4fa13ed39ae4","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051090,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[14,15],\"other_left\":[17,29],\"other_right\":[18],\"right\":[16],\"scalar\":15}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[14, 15], right=[16], another pair other_left=[17, 29], other_right=[18], and scalar=15. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[14, 15], [16]], [[31, 44], [34]], [[210, 225], [240]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[14, 15], [16]], [[31, 44], [34]], [[210, 225], [240]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0091","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52e44e7a64260399e63ab57295926c8d79279ae62091397197a5fb2947eeaa38","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051091,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[15,16],\"other_left\":[18,31],\"other_right\":[19],\"right\":[17],\"scalar\":16}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[15, 16], right=[17], another pair other_left=[18, 31], other_right=[19], and scalar=16. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[15, 16], [17]], [[33, 47], [36]], [[240, 256], [272]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[15, 16], [17]], [[33, 47], [36]], [[240, 256], [272]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0092","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d8f9882a5cfcf5730d185ddbc361d725ddcd3218b5bff226e6cbd040b5252c7","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051092,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[16,17],\"other_left\":[19,33],\"other_right\":[20],\"right\":[18],\"scalar\":17}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[16, 17], right=[18], another pair other_left=[19, 33], other_right=[20], and scalar=17. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[16, 17], [18]], [[35, 50], [38]], [[272, 289], [306]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[16, 17], [18]], [[35, 50], [38]], [[272, 289], [306]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0093","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0a8d0d5ff269972f3dbd6c8f5d120a693d5d63535fa60048d7ae27f1d9eaf77","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051093,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[17,18],\"other_left\":[20,35],\"other_right\":[21],\"right\":[19],\"scalar\":18}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[17, 18], right=[19], another pair other_left=[20, 35], other_right=[21], and scalar=18. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[17, 18], [19]], [[37, 53], [40]], [[306, 324], [342]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[17, 18], [19]], [[37, 53], [40]], [[306, 324], [342]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0094","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e02c0e30cca2f0068707bf96d1babecc5ac77fad2336084b521975e80e60ca94","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051094,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[18,19],\"other_left\":[21,37],\"other_right\":[22],\"right\":[20],\"scalar\":19}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[18, 19], right=[20], another pair other_left=[21, 37], other_right=[22], and scalar=19. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[18, 19], [20]], [[39, 56], [42]], [[342, 361], [380]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[18, 19], [20]], [[39, 56], [42]], [[342, 361], [380]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0095","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b118628a3970c31ccc52e5cf3a96f194e5222da8f084fe8ec4394e8d924af07","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051095,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[19,20],\"other_left\":[22,39],\"other_right\":[23],\"right\":[21],\"scalar\":20}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[19, 20], right=[21], another pair other_left=[22, 39], other_right=[23], and scalar=20. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[19, 20], [21]], [[41, 59], [44]], [[380, 400], [420]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[19, 20], [21]], [[41, 59], [44]], [[380, 400], [420]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0096","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"155b1dfd75cccf7e97106cf1bb494d73b5bdab8f132e6376e6ce27ea2b9fdd85","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051096,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[20,21],\"other_left\":[23,41],\"other_right\":[24],\"right\":[22],\"scalar\":21}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[20, 21], right=[22], another pair other_left=[23, 41], other_right=[24], and scalar=21. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[20, 21], [22]], [[43, 62], [46]], [[420, 441], [462]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[20, 21], [22]], [[43, 62], [46]], [[420, 441], [462]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0097","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"098b57617adc513bb786ebee32124bae6f70b26203f234087fd7f7f3085c1fe8","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051097,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[21,22],\"other_left\":[24,43],\"other_right\":[25],\"right\":[23],\"scalar\":22}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[21, 22], right=[23], another pair other_left=[24, 43], other_right=[25], and scalar=22. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[21, 22], [23]], [[45, 65], [48]], [[462, 484], [506]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[21, 22], [23]], [[45, 65], [48]], [[462, 484], [506]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0098","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e2209c5bf0701e413400dc1bbd1f4a0517f4243a778f044b8a8456d1370206e","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051098,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[22,23],\"other_left\":[25,45],\"other_right\":[26],\"right\":[24],\"scalar\":23}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[22, 23], right=[24], another pair other_left=[25, 45], other_right=[26], and scalar=23. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[22, 23], [24]], [[47, 68], [50]], [[506, 529], [552]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[22, 23], [24]], [[47, 68], [50]], [[506, 529], [552]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0099","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a927d7edac635a561a685ed04b1204cd6b61c2c88b53bc68c8a26b87695856b5","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051099,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[23,24],\"other_left\":[26,47],\"other_right\":[27],\"right\":[25],\"scalar\":24}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[23, 24], right=[25], another pair other_left=[26, 47], other_right=[27], and scalar=24. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[23, 24], [25]], [[49, 71], [52]], [[552, 576], [600]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[23, 24], [25]], [[49, 71], [52]], [[552, 576], [600]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r01-direct-sum-0100","batch_id":"batch-20261014-advance-20261105-r01-direct-sum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c72866bc70ce28425ddc830cca4d8d47dfa16f491a959cd3b016ebc2952cdb90","family_id":"qf-mod36-directsum-external-coordinate","seed":202611051100,"semantic_template":"qf-mod36-directsum-external-coordinate/{\"left\":[24,25],\"other_left\":[27,49],\"other_right\":[28],\"right\":[26],\"scalar\":25}"},"fields":{"title":"External Direct-Sum Coordinates Live on a Product Carrier","stem":"Use the pair (left,right) with left=[24, 25], right=[26], another pair other_left=[27, 49], other_right=[28], and scalar=25. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).","answer":{"kind":"expression","value":"([[24, 25], [26]], [[51, 74], [54]], [[600, 625], [650]], external_product_carrier_not_internal_subspaces)","explanation":"Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[24, 25], [26]], [[51, 74], [54]], [[600, 625], [650]], external_product_carrier_not_internal_subspaces)."},"skills":["External Direct-Sum Coordinates Live on a Product Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}