{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r02-dual-vector-space-0001","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfac9aaaa20247f34df7187bedd7deb84b013d524f4018bf77f05e5feb636576","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052001,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[1,-1],\"vector\":[2,3]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[1, -1] and primal vector=[2, 3] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([1, -1], [2, 3], -1, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([1, -1], [2, 3], -1, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0002","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a20369be6a4bf65937cbbace537992a7a0a2c2cbf4ffa4fd68564f07609e8e1a","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052002,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[2,-1],\"vector\":[2,4]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[2, -1] and primal vector=[2, 4] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([2, -1], [2, 4], 0, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([2, -1], [2, 4], 0, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0003","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9ce1865e785acdda54637adae00d8eadd85036abe782d20b2a4d1fabf968165","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052003,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[3,-1],\"vector\":[2,5]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[3, -1] and primal vector=[2, 5] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([3, -1], [2, 5], 1, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([3, -1], [2, 5], 1, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0004","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6df0581de506143228bf89113070395697ab9357ac98f869e19a471c376cb64","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052004,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[4,-1],\"vector\":[2,6]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[4, -1] and primal vector=[2, 6] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([4, -1], [2, 6], 2, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([4, -1], [2, 6], 2, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0005","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7acd4562b75d8c44b115f0a85deeee7e13c0ef9b7e401a7d089807c3693b9ef","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052005,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[5,-1],\"vector\":[2,7]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[5, -1] and primal vector=[2, 7] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([5, -1], [2, 7], 3, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([5, -1], [2, 7], 3, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0006","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c0d2b51352f69a9f6c068afaf85212466f6076540f3ecebd7daa5eb41735d4d","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052006,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[6,-1],\"vector\":[2,8]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[6, -1] and primal vector=[2, 8] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([6, -1], [2, 8], 4, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([6, -1], [2, 8], 4, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0007","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2368a4a6819e694525e3c730092877eedfa64c83a0b59c275fcb758b2adfe1e4","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052007,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[7,-1],\"vector\":[2,9]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[7, -1] and primal vector=[2, 9] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([7, -1], [2, 9], 5, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([7, -1], [2, 9], 5, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0008","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e3a770f6bf3cdf87bb28513ae9ce376decac19d47ad0839227b9ef9ca55fe06","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052008,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[8,-1],\"vector\":[2,10]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[8, -1] and primal vector=[2, 10] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([8, -1], [2, 10], 6, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([8, -1], [2, 10], 6, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0009","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d38e13d29a0f9cffe4eb5d18944f71a12cf2b89403e01ac28706edc38305d1c","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052009,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[9,-1],\"vector\":[2,11]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[9, -1] and primal vector=[2, 11] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([9, -1], [2, 11], 7, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([9, -1], [2, 11], 7, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0010","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee4d0980bf23ea87173a5d9cb9e80984be80b233b9aa9afc881888522788cf9d","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052010,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[10,-1],\"vector\":[2,12]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[10, -1] and primal vector=[2, 12] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([10, -1], [2, 12], 8, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([10, -1], [2, 12], 8, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0011","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51d986b1d5d476c431da608cd7eb988285ef7e90bac68ec86719413f22176f65","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052011,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[11,-1],\"vector\":[2,13]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[11, -1] and primal vector=[2, 13] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([11, -1], [2, 13], 9, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([11, -1], [2, 13], 9, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0012","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d94d526d51a4abf078dc3e4ace8c160d7d8190b2a25cb49790084fa92fc4468","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052012,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[12,-1],\"vector\":[2,14]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[12, -1] and primal vector=[2, 14] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([12, -1], [2, 14], 10, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([12, -1], [2, 14], 10, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0013","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"426b1f8e3f96a73827653fd8da6de681e9d13d48cf368724b8511c0c49a12351","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052013,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[13,-1],\"vector\":[2,15]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[13, -1] and primal vector=[2, 15] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([13, -1], [2, 15], 11, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([13, -1], [2, 15], 11, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0014","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5a8a9ba99b3dd8a8aafb62131d12a6f6a0770e9729e42fbdf9a043e9ae84333","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052014,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[14,-1],\"vector\":[2,16]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[14, -1] and primal vector=[2, 16] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([14, -1], [2, 16], 12, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([14, -1], [2, 16], 12, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0015","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6909cad67f003ddaa2d94821a112430d52b5698fb521583d8da918d71af6bc0a","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052015,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[15,-1],\"vector\":[2,17]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[15, -1] and primal vector=[2, 17] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([15, -1], [2, 17], 13, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([15, -1], [2, 17], 13, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0016","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c58ba1b879b9719fa0b709003e72b3196deba756e3dc94183e700deb8e859dbd","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052016,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[16,-1],\"vector\":[2,18]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[16, -1] and primal vector=[2, 18] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([16, -1], [2, 18], 14, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([16, -1], [2, 18], 14, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0017","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cbbcaaeea423b260c1892ea1a8f96ca833422d3d2569997e0d1f7e3d7a337fb","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052017,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[17,-1],\"vector\":[2,19]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[17, -1] and primal vector=[2, 19] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([17, -1], [2, 19], 15, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([17, -1], [2, 19], 15, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0018","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"487cc41b9d07c164f4e235b088225d502de075759247710d96ec394e6c4fa35a","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052018,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[18,-1],\"vector\":[2,20]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[18, -1] and primal vector=[2, 20] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([18, -1], [2, 20], 16, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([18, -1], [2, 20], 16, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0019","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d07d037ca1c02ca21b46ad8745c3935868424422df17e8ac91c8588d10c8fe63","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052019,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[19,-1],\"vector\":[2,21]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[19, -1] and primal vector=[2, 21] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([19, -1], [2, 21], 17, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([19, -1], [2, 21], 17, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0020","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29f6e3ce684020816b53d9bb21b83b19cce44027cac89fbca8268614a7dee71d","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052020,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[20,-1],\"vector\":[2,22]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[20, -1] and primal vector=[2, 22] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([20, -1], [2, 22], 18, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([20, -1], [2, 22], 18, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0021","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abab7aaec6477d44ef71930c51f7861151bfcefdb567fbb129cdc516012a05b3","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052021,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[21,-1],\"vector\":[2,23]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[21, -1] and primal vector=[2, 23] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([21, -1], [2, 23], 19, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([21, -1], [2, 23], 19, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0022","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f2212370961d2b9aabbaad2fd5093a9bc81d3624876e1f648141be5ae29349","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052022,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[22,-1],\"vector\":[2,24]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[22, -1] and primal vector=[2, 24] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([22, -1], [2, 24], 20, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([22, -1], [2, 24], 20, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0023","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e2529105cfd84d9f2573da61280da2a6db4bb622fbca55da2bdde15757a3e17","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052023,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[23,-1],\"vector\":[2,25]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[23, -1] and primal vector=[2, 25] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([23, -1], [2, 25], 21, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([23, -1], [2, 25], 21, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0024","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9dfe98c4fcef8dec46014cec043f6341681b4c7045e67357b12af50a2829ed3","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052024,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[24,-1],\"vector\":[2,26]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[24, -1] and primal vector=[2, 26] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([24, -1], [2, 26], 22, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([24, -1], [2, 26], 22, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0025","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d92f7fec34dd9bfead1dc5b932fb673e979afb39f8d203db6c161a6084d01125","family_id":"qf-mod36-dual-functional-evaluation","seed":202611052025,"semantic_template":"qf-mod36-dual-functional-evaluation/{\"coefficients\":[25,-1],\"vector\":[2,27]}"},"fields":{"title":"A Covector Evaluates a Vector but Is Not That Vector","stem":"Use row coefficients=[25, -1] and primal vector=[2, 27] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).","answer":{"kind":"expression","value":"([25, -1], [2, 27], 23, linear_functional_not_primal_vector)","explanation":"Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([25, -1], [2, 27], 23, linear_functional_not_primal_vector)."},"skills":["A Covector Evaluates a Vector but Is Not That Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0026","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b08dc5fd2d72a25ee97f3621c3f0f66a6c6c1076b68989e9e01706710dffdd8","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052026,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[1,1]],\"candidate_dual\":[[1,-1],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [1, 1]] and candidate dual rows=[[1, -1], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0027","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b81ae983da895c21e992510fa8eed7791815120907714ebb01fbbad6f750a0a8","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052027,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[2,1]],\"candidate_dual\":[[1,-2],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [2, 1]] and candidate dual rows=[[1, -2], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0028","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbbfe01e25d55fcde653da8851cd8ae44f81fde625a32a73fb3bff423008a141","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052028,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[3,1]],\"candidate_dual\":[[1,-3],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [3, 1]] and candidate dual rows=[[1, -3], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0029","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3c8be0abb046fbd56a1b993b59335a4e5201ee73397c356965d2943665694aa","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052029,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[4,1]],\"candidate_dual\":[[1,-4],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [4, 1]] and candidate dual rows=[[1, -4], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0030","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39cb0fa93a57e31b709bf8f1c4add151aa47838a87d5891db9f5c9942b328206","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052030,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[5,1]],\"candidate_dual\":[[1,-5],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [5, 1]] and candidate dual rows=[[1, -5], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0031","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4f7874cc03f8c585877eb4eb093797d0356da29399f152b9cf2673e8f44deaf","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052031,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[6,1]],\"candidate_dual\":[[1,-6],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [6, 1]] and candidate dual rows=[[1, -6], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0032","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c380f6308ef400a56ce09997c2766b2672088bf30ad4cea865f86a5f0740280","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052032,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[7,1]],\"candidate_dual\":[[1,-7],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [7, 1]] and candidate dual rows=[[1, -7], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0033","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"465468dffb6d9704f15b54c32c90114c19badaa56bf9b2c6ce3eac3e4fb56410","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052033,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[8,1]],\"candidate_dual\":[[1,-8],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [8, 1]] and candidate dual rows=[[1, -8], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0034","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"369471c415928d85b244724b10a84afb46e2a508cea91fe84e055e413727e0a4","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052034,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[9,1]],\"candidate_dual\":[[1,-9],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [9, 1]] and candidate dual rows=[[1, -9], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0035","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb2a760e5998587ddc1b1ff9ab748f04afaada74350d725e074bf9135527148c","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052035,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[10,1]],\"candidate_dual\":[[1,-10],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [10, 1]] and candidate dual rows=[[1, -10], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0036","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28f896070dceaa98d579ad7b5e1e0a131fbb055b1e46dd392fc40994efcb0c15","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052036,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[11,1]],\"candidate_dual\":[[1,-11],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [11, 1]] and candidate dual rows=[[1, -11], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0037","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"708a715dd6aaf4c06e4ff8d2c8f00961f7917acb765ba6d007e18150219e949d","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052037,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[12,1]],\"candidate_dual\":[[1,-12],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [12, 1]] and candidate dual rows=[[1, -12], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0038","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8f446d95c0dcea0f1251ad77503204ff16014efbf1e1d6dd31a8275073f1f98","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052038,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[13,1]],\"candidate_dual\":[[1,-13],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [13, 1]] and candidate dual rows=[[1, -13], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0039","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28d75bba2f2a7e7cba38d16c589005763a2a5731e081a737c7da40d8a14f13ed","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052039,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[14,1]],\"candidate_dual\":[[1,-14],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [14, 1]] and candidate dual rows=[[1, -14], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0040","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf438255baa1d410a0b90304b7364df6e2340039363179d72ed73b47ae0c6fc3","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052040,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[15,1]],\"candidate_dual\":[[1,-15],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [15, 1]] and candidate dual rows=[[1, -15], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0041","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f355202c360c7e66ceacde25daf96ed21ef0094da8416a143ed6d1578cfc5bd","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052041,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[16,1]],\"candidate_dual\":[[1,-16],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [16, 1]] and candidate dual rows=[[1, -16], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0042","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3091c9e924cf3accbbeccec87a541729752ac17893feb1f533aa33b96bb596ae","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052042,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[17,1]],\"candidate_dual\":[[1,-17],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [17, 1]] and candidate dual rows=[[1, -17], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0043","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34132185792c0a3d444a2fab36e9be47b01cea40ca594490eafcd34ec263f8bc","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052043,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[18,1]],\"candidate_dual\":[[1,-18],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [18, 1]] and candidate dual rows=[[1, -18], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0044","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52bedff328a255a30f90375866384b88a1867b64a0abd38b8080606501275273","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052044,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[19,1]],\"candidate_dual\":[[1,-19],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [19, 1]] and candidate dual rows=[[1, -19], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0045","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a240a09c6b7f09eccd10f4cd440f80b5738a6d6475dc283fdf88ee59763be38","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052045,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[20,1]],\"candidate_dual\":[[1,-20],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [20, 1]] and candidate dual rows=[[1, -20], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0046","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd6df459ed9cd8e27af8ee9bda66473f3e7b9240d6f59d27771e153116c0e664","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052046,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[21,1]],\"candidate_dual\":[[1,-21],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [21, 1]] and candidate dual rows=[[1, -21], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0047","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11e9f34d588b4c3638e46812a14520f00913f38cc9258e44f34074250b66e1c4","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052047,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[22,1]],\"candidate_dual\":[[1,-22],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [22, 1]] and candidate dual rows=[[1, -22], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0048","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d97d26aad52d191b443d341d84db8e19ad54bed821b2fbbbf079ea118cad00c","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052048,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[23,1]],\"candidate_dual\":[[1,-23],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [23, 1]] and candidate dual rows=[[1, -23], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0049","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45c057ae0d54ee89dc93984d8968bb9569bfb7d44b57c56a91fccad1dda30dde","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052049,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[24,1]],\"candidate_dual\":[[1,-24],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [24, 1]] and candidate dual rows=[[1, -24], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0050","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a83d3a3f3919fdeae9721320a893282082170daaf6daf928468d361a8e20b5f9","family_id":"qf-mod36-dual-dual-basis-certificate","seed":202611052050,"semantic_template":"qf-mod36-dual-dual-basis-certificate/{\"basis\":[[1,0],[25,1]],\"candidate_dual\":[[1,-25],[0,1]]}"},"fields":{"title":"Every Dual-Basis Pairing Must Be a Kronecker Delta","stem":"Use ordered basis column vectors=[[1, 0], [25, 1]] and candidate dual rows=[[1, -25], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)","explanation":"Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)."},"skills":["Every Dual-Basis Pairing Must Be a Kronecker Delta"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0051","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff6afbec03befc71a0b684f909368bdadbba08ed89352a7fdc648ecfd289c7c8","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052051,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":1,\"beta\":-2,\"coefficients\":[1,-2],\"u\":[1,2],\"v\":[3,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[1, -2], u=[1, 2], v=[3, -1], alpha=1, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-13, -13, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-13, -13, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0052","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dff08ca4a836430847f9e047a73a870681d44c9c8698b00f5d415268fb9e562","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052052,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":2,\"beta\":-2,\"coefficients\":[2,-2],\"u\":[1,3],\"v\":[4,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[2, -2], u=[1, 3], v=[4, -1], alpha=2, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-28, -28, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-28, -28, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0053","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"497690c9fe207c384894493c0c855657bb7245386fb8901ff799ffb71ed4f7e4","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052053,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":3,\"beta\":-2,\"coefficients\":[3,-2],\"u\":[1,4],\"v\":[5,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[3, -2], u=[1, 4], v=[5, -1], alpha=3, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-49, -49, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-49, -49, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0054","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11ae6f3c9e7715763b484c3d6fdd218f998e0571dbd55e7df340e9e3b45362f0","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052054,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":4,\"beta\":-2,\"coefficients\":[4,-2],\"u\":[1,5],\"v\":[6,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[4, -2], u=[1, 5], v=[6, -1], alpha=4, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-76, -76, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-76, -76, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0055","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdc34c237a6d110c27a4fd1e8a00320b80b9fcf7ee72ac08b086a2a7b690e153","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052055,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":5,\"beta\":-2,\"coefficients\":[5,-2],\"u\":[1,6],\"v\":[7,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[5, -2], u=[1, 6], v=[7, -1], alpha=5, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-109, -109, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-109, -109, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0056","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9917991d0832d3b24fcee3219541eb62b84eb04d3a2242efd5dd265df69c1df8","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052056,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":6,\"beta\":-2,\"coefficients\":[6,-2],\"u\":[1,7],\"v\":[8,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[6, -2], u=[1, 7], v=[8, -1], alpha=6, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-148, -148, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-148, -148, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0057","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f23debba8d71bdd83831459054b3ff55874063fa00ecd74a07feb6b132e859b","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052057,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":7,\"beta\":-2,\"coefficients\":[7,-2],\"u\":[1,8],\"v\":[9,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[7, -2], u=[1, 8], v=[9, -1], alpha=7, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-193, -193, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-193, -193, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0058","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a2b9d366726411d1d89f9a64787e955773960557f1f77dd5d31f6b6a7e4c40b","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052058,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":8,\"beta\":-2,\"coefficients\":[8,-2],\"u\":[1,9],\"v\":[10,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[8, -2], u=[1, 9], v=[10, -1], alpha=8, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-244, -244, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-244, -244, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0059","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10a86852382f2f0a674b65825bcea4e46bc575edc0a5de06ff6b21f95ff9a3a2","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052059,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":9,\"beta\":-2,\"coefficients\":[9,-2],\"u\":[1,10],\"v\":[11,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[9, -2], u=[1, 10], v=[11, -1], alpha=9, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-301, -301, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-301, -301, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0060","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30a68f82e98f7f388fc306daaaa57ec11da2818bd0272981644a854a5c66c7e1","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052060,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":10,\"beta\":-2,\"coefficients\":[10,-2],\"u\":[1,11],\"v\":[12,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[10, -2], u=[1, 11], v=[12, -1], alpha=10, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-364, -364, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-364, -364, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0061","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc3899561e3acb0c8517e4f46f77f3db87e07bc299a04537a2fd073bd61b0bc4","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052061,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":11,\"beta\":-2,\"coefficients\":[11,-2],\"u\":[1,12],\"v\":[13,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[11, -2], u=[1, 12], v=[13, -1], alpha=11, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-433, -433, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-433, -433, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0062","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6f7ac8046f4df4ec600a6a3cbec61781220776e5e9c4f7fcb842d92cc9905ae","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052062,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":12,\"beta\":-2,\"coefficients\":[12,-2],\"u\":[1,13],\"v\":[14,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[12, -2], u=[1, 13], v=[14, -1], alpha=12, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-508, -508, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-508, -508, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0063","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60ea6246bfa91b216dd04cdc51b4a71da4e1ea0df6476b171c695cd710035c67","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052063,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":13,\"beta\":-2,\"coefficients\":[13,-2],\"u\":[1,14],\"v\":[15,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[13, -2], u=[1, 14], v=[15, -1], alpha=13, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-589, -589, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-589, -589, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0064","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24ce0ab03582545fdd531389179a5d74b99415bfb0920a08476d34b3d25a08f4","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052064,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":14,\"beta\":-2,\"coefficients\":[14,-2],\"u\":[1,15],\"v\":[16,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[14, -2], u=[1, 15], v=[16, -1], alpha=14, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-676, -676, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-676, -676, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0065","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3436f902d03a45dc165f4ab5fd43507efd9c24099d274eded117f361e94cb074","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052065,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":15,\"beta\":-2,\"coefficients\":[15,-2],\"u\":[1,16],\"v\":[17,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[15, -2], u=[1, 16], v=[17, -1], alpha=15, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-769, -769, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-769, -769, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0066","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ac0824a3dada13d18e962517f6979ba55af643238c45a58a9ce087d231b3061","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052066,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":16,\"beta\":-2,\"coefficients\":[16,-2],\"u\":[1,17],\"v\":[18,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[16, -2], u=[1, 17], v=[18, -1], alpha=16, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-868, -868, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-868, -868, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0067","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b95415377478165855c881d962144fd91a5989512f632825f8fb2fef7acaab71","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052067,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":17,\"beta\":-2,\"coefficients\":[17,-2],\"u\":[1,18],\"v\":[19,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[17, -2], u=[1, 18], v=[19, -1], alpha=17, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-973, -973, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-973, -973, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0068","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c2f067dee25777e74a814c10ae5734d5864b880404c33fa7889695471aa69c4","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052068,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":18,\"beta\":-2,\"coefficients\":[18,-2],\"u\":[1,19],\"v\":[20,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[18, -2], u=[1, 19], v=[20, -1], alpha=18, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-1084, -1084, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1084, -1084, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0069","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"810fd46644adb11af92bcb1f193a49dd7d0c35331674dc979bb6d025de35dacd","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052069,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":19,\"beta\":-2,\"coefficients\":[19,-2],\"u\":[1,20],\"v\":[21,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[19, -2], u=[1, 20], v=[21, -1], alpha=19, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-1201, -1201, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1201, -1201, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0070","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5ce36b7c71ecc43fd683123480d2d7398cab71b7cc2a1a4f9dd3e295db24253","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052070,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":20,\"beta\":-2,\"coefficients\":[20,-2],\"u\":[1,21],\"v\":[22,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[20, -2], u=[1, 21], v=[22, -1], alpha=20, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-1324, -1324, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1324, -1324, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0071","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fadd7aea8a41a92ac346ba005a275c780815d9a8a92f3edb8343fe7cd841cee","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052071,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":21,\"beta\":-2,\"coefficients\":[21,-2],\"u\":[1,22],\"v\":[23,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[21, -2], u=[1, 22], v=[23, -1], alpha=21, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-1453, -1453, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1453, -1453, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0072","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"444c718e777f3cef4d291057311504e84ab91eade4fba6e21512a4c901304341","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052072,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":22,\"beta\":-2,\"coefficients\":[22,-2],\"u\":[1,23],\"v\":[24,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[22, -2], u=[1, 23], v=[24, -1], alpha=22, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-1588, -1588, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1588, -1588, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0073","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc15f10adf6fbf2fcdef95f7340ee715ce5ea6d28322d6752b717ed3acc3f60c","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052073,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":23,\"beta\":-2,\"coefficients\":[23,-2],\"u\":[1,24],\"v\":[25,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[23, -2], u=[1, 24], v=[25, -1], alpha=23, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-1729, -1729, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1729, -1729, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0074","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd782f6827e39b8158429c99b33271c68a8d9d6fc4f0757361eef83399a9c6bf","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052074,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":24,\"beta\":-2,\"coefficients\":[24,-2],\"u\":[1,25],\"v\":[26,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[24, -2], u=[1, 25], v=[26, -1], alpha=24, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-1876, -1876, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1876, -1876, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0075","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ab3c45d036eafabb471472b1cad7561b1e6fc1042c372be1fe23b6be059a980","family_id":"qf-mod36-dual-linearity-certificate","seed":202611052075,"semantic_template":"qf-mod36-dual-linearity-certificate/{\"alpha\":25,\"beta\":-2,\"coefficients\":[25,-2],\"u\":[1,26],\"v\":[27,-1]}"},"fields":{"title":"One Row Formula Supplies a Symbolic Linearity Identity","stem":"Use covector row=[25, -2], u=[1, 26], v=[27, -1], alpha=25, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).","answer":{"kind":"expression","value":"(-2029, -2029, Yes, row_covector_gives_universal_linearity_certificate)","explanation":"Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-2029, -2029, Yes, row_covector_gives_universal_linearity_certificate)."},"skills":["One Row Formula Supplies a Symbolic Linearity Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0076","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62d22c6b724f0b82d7637d3e725efd449ac2a8baeb3003523309a6dfb2fac11b","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052076,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,1],[0,1]],\"covector_coordinates\":[2,1],\"primal_coordinates\":[2,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 1], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[2, 3] and old covector row=[2, 1]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-1, 3], [2, 3], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-1, 3], [2, 3], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0077","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa09f8e7b5c89857f2759ba8395849b3960400082a0f679c1c72a79168a4b41d","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052077,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,2],[0,1]],\"covector_coordinates\":[2,2],\"primal_coordinates\":[3,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 2], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[3, 3] and old covector row=[2, 2]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-3, 3], [2, 6], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-3, 3], [2, 6], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0078","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9156c1d8736a1b4d41e7027d43db91d3846133d9ba3c2e2a5ce14d2ae123b731","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052078,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,3],[0,1]],\"covector_coordinates\":[2,3],\"primal_coordinates\":[4,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 3], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[4, 3] and old covector row=[2, 3]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-5, 3], [2, 9], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-5, 3], [2, 9], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0079","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2086cbca89ded2b4cd77e18e02014f1aa7be9a090d59df49a0b518999b6b85b7","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052079,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,4],[0,1]],\"covector_coordinates\":[2,4],\"primal_coordinates\":[5,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 4], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[5, 3] and old covector row=[2, 4]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-7, 3], [2, 12], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-7, 3], [2, 12], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0080","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24661215c679783183537ccd4f383b90d8b1e6baea423d44898fca9cf0e86254","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052080,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,5],[0,1]],\"covector_coordinates\":[2,5],\"primal_coordinates\":[6,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 5], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[6, 3] and old covector row=[2, 5]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-9, 3], [2, 15], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-9, 3], [2, 15], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0081","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04634b7bc2739ba01588f56a28753931a2c769143bc4c445f6e744c524414324","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052081,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,6],[0,1]],\"covector_coordinates\":[2,6],\"primal_coordinates\":[7,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 6], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[7, 3] and old covector row=[2, 6]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-11, 3], [2, 18], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-11, 3], [2, 18], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0082","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eca23a629a68dadc67d3900c9db96a1d2d2c338c1ba188e3a6e3c121b23ef206","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052082,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,7],[0,1]],\"covector_coordinates\":[2,7],\"primal_coordinates\":[8,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 7], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[8, 3] and old covector row=[2, 7]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-13, 3], [2, 21], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-13, 3], [2, 21], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0083","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66c7043da6a1686aa50b8f48d6eee3d86eafb4a640d2597d2f27ef475bd107e7","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052083,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,8],[0,1]],\"covector_coordinates\":[2,8],\"primal_coordinates\":[9,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 8], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[9, 3] and old covector row=[2, 8]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-15, 3], [2, 24], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-15, 3], [2, 24], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0084","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b18d2171c3a86274614487dbe4affcbf79ebd7628d82a89f45b95ace19743310","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052084,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,9],[0,1]],\"covector_coordinates\":[2,9],\"primal_coordinates\":[10,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 9], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[10, 3] and old covector row=[2, 9]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-17, 3], [2, 27], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-17, 3], [2, 27], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0085","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9803d4a6e2ee640bf5d8cb51fc0d2f8ee251949a6f0fd745960a3a1bb592c8f7","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052085,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,10],[0,1]],\"covector_coordinates\":[2,10],\"primal_coordinates\":[11,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 10], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[11, 3] and old covector row=[2, 10]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-19, 3], [2, 30], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-19, 3], [2, 30], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0086","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5113677a5811d78e233a3583643f3d664ee2fed83f725e1be0f318a2fd35d6d","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052086,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,11],[0,1]],\"covector_coordinates\":[2,11],\"primal_coordinates\":[12,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 11], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[12, 3] and old covector row=[2, 11]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-21, 3], [2, 33], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-21, 3], [2, 33], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0087","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84366badfcdd153f14d7baf35e0079742dad42d4d6f7564678e15e3be9c70586","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052087,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,12],[0,1]],\"covector_coordinates\":[2,12],\"primal_coordinates\":[13,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 12], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[13, 3] and old covector row=[2, 12]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-23, 3], [2, 36], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-23, 3], [2, 36], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0088","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78e4933c936188a6c3f547ed92164b5868d451f5e2550f3b617402bedfbeed24","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052088,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,13],[0,1]],\"covector_coordinates\":[2,13],\"primal_coordinates\":[14,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 13], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[14, 3] and old covector row=[2, 13]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-25, 3], [2, 39], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-25, 3], [2, 39], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0089","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a6ee70dedcb77efd060a0279579f46ff5dff8b3f19bb3b26c3b5dee74d27925","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052089,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,14],[0,1]],\"covector_coordinates\":[2,14],\"primal_coordinates\":[15,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 14], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[15, 3] and old covector row=[2, 14]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-27, 3], [2, 42], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-27, 3], [2, 42], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0090","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67cb904d23e3f97f3ccc1b98e8088cdddaba1a11568801f8b86c1485d4b4554f","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052090,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,15],[0,1]],\"covector_coordinates\":[2,15],\"primal_coordinates\":[16,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 15], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[16, 3] and old covector row=[2, 15]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-29, 3], [2, 45], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-29, 3], [2, 45], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0091","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a14c63b39c4d5ca7f4e901b37c27b96e98e1b6176ee79d5706d95e0951163042","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052091,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,16],[0,1]],\"covector_coordinates\":[2,16],\"primal_coordinates\":[17,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 16], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[17, 3] and old covector row=[2, 16]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-31, 3], [2, 48], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-31, 3], [2, 48], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0092","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9cdc9831a7976052e2599e6ee187756763a72bdca35d39a0188bd4ef07f83ac","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052092,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,17],[0,1]],\"covector_coordinates\":[2,17],\"primal_coordinates\":[18,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 17], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[18, 3] and old covector row=[2, 17]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-33, 3], [2, 51], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-33, 3], [2, 51], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0093","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f29781e383d2f07d2d78907d705b7f25d85c2c93ef96a82b1390916ec1c37d01","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052093,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,18],[0,1]],\"covector_coordinates\":[2,18],\"primal_coordinates\":[19,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 18], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[19, 3] and old covector row=[2, 18]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-35, 3], [2, 54], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-35, 3], [2, 54], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0094","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19d3cc3f9b31651bc4073c03ad8a646630baeb54454337a37c1f651184a76fbe","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052094,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,19],[0,1]],\"covector_coordinates\":[2,19],\"primal_coordinates\":[20,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 19], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[20, 3] and old covector row=[2, 19]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-37, 3], [2, 57], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-37, 3], [2, 57], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0095","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6529aaefcbb6002c963e25210724cf24c7d104af7cec24e7dd2c1cabda44e8c0","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052095,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,20],[0,1]],\"covector_coordinates\":[2,20],\"primal_coordinates\":[21,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 20], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[21, 3] and old covector row=[2, 20]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-39, 3], [2, 60], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-39, 3], [2, 60], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0096","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"521546145aab0d74556f8ea166060e2d266b22c984a51ae7e22d70b9751eb863","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052096,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,21],[0,1]],\"covector_coordinates\":[2,21],\"primal_coordinates\":[22,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 21], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[22, 3] and old covector row=[2, 21]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-41, 3], [2, 63], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-41, 3], [2, 63], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0097","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4af0d2e24b64cc797cd775f477b7fa4626a8eaaa4c6ad2a123baf9fb8e0da5e9","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052097,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,22],[0,1]],\"covector_coordinates\":[2,22],\"primal_coordinates\":[23,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 22], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[23, 3] and old covector row=[2, 22]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-43, 3], [2, 66], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-43, 3], [2, 66], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0098","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb235dcaffc135f01b48ea037ce57abc70151aa3a0a70f98b95b617727161d8c","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052098,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,23],[0,1]],\"covector_coordinates\":[2,23],\"primal_coordinates\":[24,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 23], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[24, 3] and old covector row=[2, 23]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-45, 3], [2, 69], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-45, 3], [2, 69], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0099","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81f42f62fc0e04bec3169e75e5b21897989aef1d193629f93ee949f7715803b5","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052099,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,24],[0,1]],\"covector_coordinates\":[2,24],\"primal_coordinates\":[25,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 24], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[25, 3] and old covector row=[2, 24]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-47, 3], [2, 72], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-47, 3], [2, 72], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions 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-r02-dual-vector-space-0100","batch_id":"batch-20261014-advance-20261105-r02-dual-vector-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc8453f665c8ea15ffa2567593e12fdce4b9dc2930c59a6ba1149f6ea7ba6bf5","family_id":"qf-mod36-dual-coordinate-change-boundary","seed":202611052100,"semantic_template":"qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,25],[0,1]],\"covector_coordinates\":[2,25],\"primal_coordinates\":[26,3]}"},"fields":{"title":"Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws","stem":"Use invertible change matrix P=[[1, 25], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[26, 3] and old covector row=[2, 25]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).","answer":{"kind":"expression","value":"([-49, 3], [2, 75], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)","explanation":"Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-49, 3], [2, 75], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)."},"skills":["Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}