{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0001","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b94351548db7e92bdce63983c9baabc1d3f03891f86c9353c225e7f25200872","family_id":"qf-mod42-higher-order-qualification","seed":202611112001,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[1,2,3,4],\"order\":0}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(3, 0, Yes, f_zero_equals_f, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 0, Yes, f_zero_equals_f, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0002","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c37d95495fc75fc0bfe06fa499b10f1b9be1e2024d0773601e511c3e1df4bb8","family_id":"qf-mod42-higher-order-qualification","seed":202611112002,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[2,4,6,8,10],\"order\":1}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(4, 1, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 1, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0003","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e171ec0c28bbe526acb326382605493ba27748dd59f32a998bfb01b1dc54949","family_id":"qf-mod42-higher-order-qualification","seed":202611112003,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[3,6,9,12,15,18],\"order\":2}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(5, 2, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 2, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0004","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a07c2ca54b18f7ab09c9912692df802f697c5a1694a5fce41ca06791a22c8ee","family_id":"qf-mod42-higher-order-qualification","seed":202611112004,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[4,8,12,16,20,24,28],\"order\":3}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(6, 3, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0005","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a393cce799248c93bc5588cf55df31b324742d7519bb69b006ef0b98f9c533c1","family_id":"qf-mod42-higher-order-qualification","seed":202611112005,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[5,10,15,20],\"order\":4}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(3, 4, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 4, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0006","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b68e37f3f4ef78e4be1698d5cfc1c16c25af1ea69ded33760dee797fd5ad3f8","family_id":"qf-mod42-higher-order-qualification","seed":202611112006,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[6,12,18,24,30],\"order\":5}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(4, 5, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 5, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0007","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67724b6d329c3c8aa85fe63212ff74de38477f1cea6f92d7fb959f61662bec2b","family_id":"qf-mod42-higher-order-qualification","seed":202611112007,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[7,14,21,28,35,42],\"order\":6}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(5, 6, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 6, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0008","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6775aba27fcdfe822d74aaae61f24f18bef1222dca0939f8e064ec662704593","family_id":"qf-mod42-higher-order-qualification","seed":202611112008,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[8,16,24,32,40,48,56],\"order\":7}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(6, 7, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0009","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"770fbc209d43f593085c7dc3db9dd2de7655e8b7068325db29e46a30c16f1d4c","family_id":"qf-mod42-higher-order-qualification","seed":202611112009,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[9,18,27,36],\"order\":3}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(3, 3, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 3, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0010","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77ad952797776f4cdd2db0d7c08d9f3b1075ff94bb08b6c3d158ae54c394b2e8","family_id":"qf-mod42-higher-order-qualification","seed":202611112010,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[10,20,30,40,50],\"order\":3}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(4, 3, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 3, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0011","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7137d09bb7190c1d5f421d6183ff9cb01e02ca69a1fe5f69769beff1d41d200c","family_id":"qf-mod42-higher-order-qualification","seed":202611112011,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[11,22,33,44,55,66],\"order\":3}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(5, 3, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 3, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0012","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c08acaaa6d27b70fab1ada4cb48889c6b0caab3eb5ba757c25664eafa940ccf8","family_id":"qf-mod42-higher-order-qualification","seed":202611112012,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[12,24,36,48,60,72,84],\"order\":3}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(6, 3, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0013","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7c1437c9905060da3d4188c0e154ae6326d3bd53c11e43ab53eb6e915bcad1d","family_id":"qf-mod42-higher-order-qualification","seed":202611112013,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[13,26,39,52],\"order\":2}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(3, 2, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 2, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0014","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34ee63815993c592747b2685ea61a535414be2d99cd5d8b558b7f57dad01ac89","family_id":"qf-mod42-higher-order-qualification","seed":202611112014,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[14,28,42,56,70],\"order\":1}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(4, 1, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 1, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0015","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebf3a54a2445392747462277ff791bc06ff5cf7a68a90717ba17287cd71feacb","family_id":"qf-mod42-higher-order-qualification","seed":202611112015,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[15,30,45,60,75,90],\"order\":0}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(5, 0, Yes, f_zero_equals_f, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 0, Yes, f_zero_equals_f, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0016","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15c80a004c02be4d5c0ce7dec54ce33e354a98a2d03b794954357f2c41422c69","family_id":"qf-mod42-higher-order-qualification","seed":202611112016,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[16,32,48,64,80,96,112],\"order\":7}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(6, 7, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0017","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab6b1c444c6f698b9667d5dc75e198d001532302da5bd71ed5e191bec3d114d","family_id":"qf-mod42-higher-order-qualification","seed":202611112017,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[17,34,51,68],\"order\":1}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(3, 1, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 1, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0018","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60b3e05aa7bf07a8efeb2f8bda5efe12376d1fc775725a7228cc5124b8f76b22","family_id":"qf-mod42-higher-order-qualification","seed":202611112018,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[18,36,54,72,90],\"order\":5}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(4, 5, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 5, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0019","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"922fffdd415fa787ea33343b93389b737c886a5f8a445ec6741d5c39e8795084","family_id":"qf-mod42-higher-order-qualification","seed":202611112019,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[19,38,57,76,95,114],\"order\":4}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(5, 4, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 4, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0020","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae86e2b071b4719ebd03615d720c75fc0cbedd16702095ef2016dec0ef01da53","family_id":"qf-mod42-higher-order-qualification","seed":202611112020,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[20,40,60,80,100,120,140],\"order\":3}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(6, 3, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0021","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d942ab97e384ea8a9b9a9597e31e77c9d10ed541204c2c93b139984832dadaa","family_id":"qf-mod42-higher-order-qualification","seed":202611112021,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[21,42,63,84],\"order\":0}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(3, 0, Yes, f_zero_equals_f, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 0, Yes, f_zero_equals_f, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0022","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f4140244c27f81ff6916230180c114eaa2e2d7e0cf6344cc0d9dc5d29cc6e79","family_id":"qf-mod42-higher-order-qualification","seed":202611112022,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[22,44,66,88,110],\"order\":3}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(4, 3, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 3, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0023","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec1184102ea85b0db84dd2bbfa6dc5b9f65777e770319f8f4f47d51980dba354","family_id":"qf-mod42-higher-order-qualification","seed":202611112023,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[23,46,69,92,115,138],\"order\":1}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(5, 1, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 1, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0024","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0151c7576004277e1bc3db5d5cfe625b78d27a895b0529996833ce5e4edc0371","family_id":"qf-mod42-higher-order-qualification","seed":202611112024,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[24,48,72,96,120,144,168],\"order\":7}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(6, 7, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0025","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbd7439b9513c1d0cfde7925a45936db415bbcd6f5761acd161dccc9203a43b5","family_id":"qf-mod42-higher-order-qualification","seed":202611112025,"semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[25,50,75,100],\"order\":4}"},"fields":{"title":"Order Zero Is the Original Function; Later Orders Need the Recursive Chain","stem":"Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).","answer":{"kind":"expression","value":"(3, 4, Yes, recursive_derivative_chain_required, Yes)","explanation":"Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 4, Yes, recursive_derivative_chain_required, Yes)."},"skills":["Order Zero Is the Original Function; Later Orders Need the Recursive Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0026","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37f6ebe85cd282eb796672278f5adcf97053070c88f1127075691088b5982131","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112026,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[1,2,3,4],\"order\":0,\"point\":-2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([1, 2, 3, 4], 0, [1, 2, 3, 4], -2, -23, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([1, 2, 3, 4], 0, [1, 2, 3, 4], -2, -23, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0027","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffd5a006f8f7faee0be23d39505d0dcbfeee25f5202df6de71d9d83ab8d37b25","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112027,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[2,4,6,8,10],\"order\":1,\"point\":-1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([2, 4, 6, 8, 10], 1, [4, 12, 24, 40], -1, -24, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([2, 4, 6, 8, 10], 1, [4, 12, 24, 40], -1, -24, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0028","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae4b592c250c4c3c3880df4c053f4a426df12e3f70d78916efa057a3e03b4109","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112028,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[3,6,9,12,15,18],\"order\":2,\"point\":0}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0029","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a88bbad6961a1db8cf0aa0fea6bd05753090c55986ebac77e536a576b76c170","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112029,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[4,8,12,16,20,24,28],\"order\":3,\"point\":1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0030","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a325a2aa21ae9438114371ab8ba3d5e62549dd89c1c51dbd45cf60706a3a0ef","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112030,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[5,10,15,20],\"order\":4,\"point\":2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([5, 10, 15, 20], 4, [0], 2, 0, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([5, 10, 15, 20], 4, [0], 2, 0, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0031","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d57d11a826b207f12c206bba61d5ec8ff9f56c362c56b6699d27a1c44b85735a","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112031,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[6,12,18,24,30],\"order\":5,\"point\":-2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([6, 12, 18, 24, 30], 5, [0], -2, 0, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([6, 12, 18, 24, 30], 5, [0], -2, 0, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0032","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"743011eaa633f8f9d7321943cffade48d3a9ef42d85dcbd1517df4b4f3461cf9","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112032,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[7,14,21,28,35,42],\"order\":6,\"point\":-1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0033","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6c51779bc128212f198b38c0dd4e0388ded49eac9fa38dc221b2a60df60d9f6","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112033,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[8,16,24,32,40,48,56],\"order\":7,\"point\":0}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([8, 16, 24, 32, 40, 48, 56], 7, [0], 0, 0, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([8, 16, 24, 32, 40, 48, 56], 7, [0], 0, 0, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0034","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f2a38ef8f37e495be76dc8da0601f9df5a66000e64b446edfeb8a7779565afb","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112034,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[9,18,27,36],\"order\":3,\"point\":1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([9, 18, 27, 36], 3, [216], 1, 216, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([9, 18, 27, 36], 3, [216], 1, 216, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0035","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac7f1669f0c89af62e1eecaa7914f48bdb5aa145557b0535c53e17ae7eff1403","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112035,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[10,20,30,40,50],\"order\":3,\"point\":2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0036","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bd127c4d9976ff59921ec67ba4bbea2675b9901fbe0fd86857a512a236339ec","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112036,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[11,22,33,44,55,66],\"order\":3,\"point\":-2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0037","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d89bf8ceb72652e552ae212fc87410d9c9fc33917c3259ee8bf5aa5330660d47","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112037,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[12,24,36,48,60,72,84],\"order\":3,\"point\":-1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([12, 24, 36, 48, 60, 72, 84], 3, [288, 1440, 4320, 10080], -1, -6912, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([12, 24, 36, 48, 60, 72, 84], 3, [288, 1440, 4320, 10080], -1, -6912, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0038","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bb3676d985550926818e000f5976ea307d6905607feaab6d5d9c81d66b4bc65","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112038,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[13,26,39,52],\"order\":2,\"point\":0}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([13, 26, 39, 52], 2, [78, 312], 0, 78, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([13, 26, 39, 52], 2, [78, 312], 0, 78, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0039","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67ec98660f3346a335039457f2ffeecbf4990816d0845091d6bc50c307de6728","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112039,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[14,28,42,56,70],\"order\":1,\"point\":1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0040","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd898a7f129fe4461a3bfee00174e6f1f935440b9388129a8187d94a74bbd0c1","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112040,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[15,30,45,60,75,90],\"order\":0,\"point\":2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0041","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4de15f92aa3c0b467ff9bfc96224a3134c99cc2b47071e84434a67bdea72b106","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112041,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0042","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a220acab71fcf3f3c440bcd5cdf7afe5d16b3feb480f2e3660794d138df204e","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112042,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[17,34,51,68],\"order\":1,\"point\":-1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([17, 34, 51, 68], 1, [34, 102, 204], -1, 136, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([17, 34, 51, 68], 1, [34, 102, 204], -1, 136, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0043","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0cf8fa6695b31b2af30018a467e1c648fbd78b3965b769c7d67ae7db8dce9d7","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112043,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[18,36,54,72,90],\"order\":5,\"point\":0}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([18, 36, 54, 72, 90], 5, [0], 0, 0, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([18, 36, 54, 72, 90], 5, [0], 0, 0, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0044","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d5cd4f1444bddaf7b6ab51c15cde85fa2020b1523f3b371a2dd31a5e158c5d2","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112044,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[19,38,57,76,95,114],\"order\":4,\"point\":1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0045","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35a2730433950dd1dee1ebe144fa8762fc29987d2666e5a2bfcd0274af079991","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112045,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[20,40,60,80,100,120,140],\"order\":3,\"point\":2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([20, 40, 60, 80, 100, 120, 140], 3, [480, 2400, 7200, 16800], 2, 168480, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([20, 40, 60, 80, 100, 120, 140], 3, [480, 2400, 7200, 16800], 2, 168480, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0046","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6373a14c3707d6511135b00912417b939c715a076b5dc33f8dd5083e6d9f84dc","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112046,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[21,42,63,84],\"order\":0,\"point\":-2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0047","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08e4591cd797ee5561ecbf44bd3fcee2198535bd4a183044c7cd07c45a2268ae","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112047,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[22,44,66,88,110],\"order\":3,\"point\":-1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([22, 44, 66, 88, 110], 3, [528, 2640], -1, -2112, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([22, 44, 66, 88, 110], 3, [528, 2640], -1, -2112, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0048","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc1ddd0f70fac6e42475886b8ca22d9ab9b09c45babb5e6e43a2981fdcb3755a","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112048,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[23,46,69,92,115,138],\"order\":1,\"point\":0}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([23, 46, 69, 92, 115, 138], 1, [46, 138, 276, 460, 690], 0, 46, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([23, 46, 69, 92, 115, 138], 1, [46, 138, 276, 460, 690], 0, 46, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0049","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5c88466f8f607fda4ce202d3c97e8fe36666c5c20472bf1ed0920003b56f719","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112049,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[24,48,72,96,120,144,168],\"order\":7,\"point\":1}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([24, 48, 72, 96, 120, 144, 168], 7, [0], 1, 0, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([24, 48, 72, 96, 120, 144, 168], 7, [0], 1, 0, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0050","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a082fdcf04b645dd8deaffc61ac4b0faa467a2ee2d0a48acee0dace061f1b97","family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112050,"semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[25,50,75,100],\"order\":4,\"point\":2}"},"fields":{"title":"Exact Falling Factorials Preserve Low-to-High Derivative Coefficients","stem":"Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).","answer":{"kind":"expression","value":"([25, 50, 75, 100], 4, [0], 2, 0, falling_factorial_coefficients_exact)","explanation":"Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([25, 50, 75, 100], 4, [0], 2, 0, falling_factorial_coefficients_exact)."},"skills":["Exact Falling Factorials Preserve Low-to-High Derivative Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0051","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bee48e6b7ca13cffe139f012407220c8a15043955a76e9b31b62821071c70abe","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112051,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[1,2,3,4],\"order\":4,\"point\":-2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(3, 4, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0052","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bdbec24967cac3b6f44d74f627501d65f3d67ebaba290b8060ec6a44d588fb1","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112052,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[2,4,6,8,10],\"order\":6,\"point\":-1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(4, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0053","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"184f94c5386532f60151920c53dd04f9a4c2e0755ef7befa151c743cf6543c6c","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112053,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[3,6,9,12,15,18],\"order\":8,\"point\":0}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(5, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0054","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb0ec1f96202d958221009a5fa377586867ed9b9a4f805c460dfc1e3d6036f52","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112054,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[4,8,12,16,20,24,28],\"order\":7,\"point\":1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(6, 7, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 7, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0055","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83e7bb22f87244cadf60c0baa58b43d986514bd49cf0d55c51a3b06bb6c1549f","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112055,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[5,10,15,20],\"order\":5,\"point\":2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(3, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0056","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b82f6858baf6d1f4416ea764830f21912b6561988e4ac1eb4a2f81ec08910f12","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112056,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[6,12,18,24,30],\"order\":7,\"point\":-2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(4, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0057","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7fd98201f94537dde2375d55d6cf66f0ce6031977ab0ebd011687dcc18b7bc4","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112057,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[7,14,21,28,35,42],\"order\":6,\"point\":-1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(5, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0058","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"603cd2f43a5416d1854c5962de9f882f3b206ca117cb1989dbfc6e88f4508044","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112058,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[8,16,24,32,40,48,56],\"order\":8,\"point\":0}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(6, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0059","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e6f62646116d85d31afdf0fc3d18cf7ea15bfd6764ddd6c7d768e9bb77266ca","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112059,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[9,18,27,36],\"order\":6,\"point\":1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(3, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0060","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be7d05f8e342578d18b36ced3d70ee2badef57da8bf0c25e57f8a3cd285e8afe","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112060,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[10,20,30,40,50],\"order\":5,\"point\":2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(4, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0061","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e659edcb89ae01b607ffe92365a2fcdd6d9e02ceb86a003781004e7f798ac989","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112061,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[11,22,33,44,55,66],\"order\":7,\"point\":-2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(5, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0062","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b677432ba139740c54d0588306ee6509dac6425c8dbf986177d289975a369db3","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112062,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[12,24,36,48,60,72,84],\"order\":9,\"point\":-1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=9≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(6, 9, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 9, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0063","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e285043928b9fdf2e5bed6f18752d4c2d0d2cdc0822a87dc4248aeb9f7b6f06","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112063,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[13,26,39,52],\"order\":4,\"point\":0}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(3, 4, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0064","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"084f5dd4a816974cdf700384a1ab449aa3919a54cd527be464400ab6b60708ae","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112064,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[14,28,42,56,70],\"order\":6,\"point\":1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(4, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0065","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03358e8d3c257d25052fddb0eb854f6ab2ea15b98cd64bae27f9e4987d134384","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112065,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[15,30,45,60,75,90],\"order\":8,\"point\":2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(5, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0066","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbc5480e2de7c09b225df53de6902ef907b9c7344b1b7f7afa565b2a1cdf6d0d","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112066,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(6, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0067","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"091dbc78c1869a89c94404849a8e60b341c21f328d0099bc8e331e290df32853","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112067,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[17,34,51,68],\"order\":5,\"point\":-1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(3, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0068","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5e83e02c451426956a01a72f70318de8be200c601a72110788b01b745e41620","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112068,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[18,36,54,72,90],\"order\":7,\"point\":0}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(4, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0069","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92ec45067d4a93ef5aabd64a79f2420691b0cf1e9bc32f1a24100f6712c87780","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112069,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[19,38,57,76,95,114],\"order\":6,\"point\":1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(5, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0070","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01842726b1895afac04f17d860c1f72b2d4fff891eb5ea7599e18439386336ce","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112070,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[20,40,60,80,100,120,140],\"order\":8,\"point\":2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(6, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0071","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fe8567f4bdc3bc379f4a997a1e93f5e088287a91970f24f5125bae8b8ed2dae","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112071,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[21,42,63,84],\"order\":6,\"point\":-2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(3, 6, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 6, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0072","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43e17fd85a17a9471bc20484247e5ff678500d679d30ace613af9d1549437b69","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112072,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[22,44,66,88,110],\"order\":5,\"point\":-1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(4, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0073","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f38f4496b97054405eab3db5625ca80601a057663a152c42f5c41c60bfff0df","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112073,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[23,46,69,92,115,138],\"order\":7,\"point\":0}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(5, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0074","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36442dd152071a58d757f699d7e6a77003766f92dc83c096ee1c9e8a109f2183","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112074,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[24,48,72,96,120,144,168],\"order\":9,\"point\":1}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=9≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(6, 9, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 9, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0075","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe4963bc208445e26fe969606f27abc9a399739a98dd8ef61a1e11dc0772e1bc","family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112075,"semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[25,50,75,100],\"order\":4,\"point\":2}"},"fields":{"title":"Orders above the Declared Polynomial Degree Bound Give Zero","stem":"Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).","answer":{"kind":"expression","value":"(3, 4, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation":"Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)."},"skills":["Orders above the Declared Polynomial Degree Bound Give Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0076","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7ef760734bd05659a52cd1d7cb8142c94b3bdcee1b1f648fb7cf44052694a13","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112076,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":1}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=1 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(1, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (1, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0077","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a0d1f15aafa1ef2e7333ba8ccb30f3d16d01f6a33cc5625a39fe43de69a4537","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112077,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":2}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=2 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(2, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (2, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0078","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4a40b57841852f122cfe14179b9bbfdb83c542cc3cd640924db804924253364","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112078,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":3}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=3 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(3, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (3, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0079","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c884b70a73253091b1f252d72438b3db590820b504f54e71018b0eee0b6156a","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112079,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":4}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=4 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(4, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (4, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0080","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cac90aa7cdb12b048b4a4a76dcb3bcd6c0ebb0e49010d85a09a06538edee7586","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112080,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":5}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=5 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(5, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (5, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0081","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5b1e1556b35b0cebfb3f75109bc5f9e41c0fc05ca57200851ca17e2b2be29b9","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112081,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":6}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=6 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(6, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (6, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0082","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"effd305825b9dd90576263443c395f4bd165a32a2b4b8dc3a6a8a477d29057b4","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112082,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":7}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=7 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(7, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (7, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0083","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4dea189a426086f0ed865a2ca7f4add2f52fd7e835f58f57f5fe185abf9e86d","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112083,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":8}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=8 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(8, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (8, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0084","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca4d2073d1e6569644c3fc254c18f0c66b0ad9a9c86746dcd5d368a3d31cbcfc","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112084,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":9}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=9 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(9, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (9, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0085","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e39e2bd53c778498539d342e4ceff1c6dc8f5db6118beb063d1984644e7c1ac","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112085,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":10}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=10 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(10, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (10, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0086","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14af5e7df6af389920f21ac0add79bb62ae4a66ded6a18cc6f44c7437f0a04a3","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112086,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":11}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=11 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(11, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (11, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0087","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c6e0697637b07e60904bcb671fd0a21aa2a0f09402a8c6e48f0b5173ac70b0","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112087,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":12}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=12 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(12, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (12, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0088","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d60866dbbd66bf4b49f3afd3d425b9dcb37cf8739bf6c10327e925f56fda9d8","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112088,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":13}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=13 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(13, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (13, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0089","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99f02acc3b640a6a99028e7c1619d852f30d8f894f993ff05666769f09242a52","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112089,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":14}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=14 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(14, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (14, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0090","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c2bf58c054be94f122b5c5e82fbbc7016953c2082165dd672f67a7a8216a257","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112090,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":15}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=15 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(15, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (15, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0091","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36976187e85c3bb3aa511ec2eb14512bdd57f637bbd026a79e23bf80bd4360f6","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112091,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":16}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=16 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(16, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (16, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0092","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47f2e7c7179c9fa44db09f172eb2299f0919163e2c59d18203e506a113c56e48","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112092,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":17}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=17 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(17, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (17, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0093","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5efd3c4b0b3030b5930e8d41e40a72910441cc12ae75bf9f7f00052008eb910b","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112093,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":18}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=18 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(18, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (18, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0094","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3549da2be2562ca5c1cba3c232a4ecf983b8e5615a8dcbb4451205631260da83","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112094,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":19}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=19 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(19, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (19, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0095","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"822e4f89d4944bd95ebaf06ac44c11edb792e08342f3e48923b7b161d9070328","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112095,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":20}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=20 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(20, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (20, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0096","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"278c95753d6106582d5ebae66f2fe54cdadb54a8a01dfa444a64516d3d485aab","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112096,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":21}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=21 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(21, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (21, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0097","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdb769b8fc36335f549a9157b1175a5087130dfba8ac982941c86b2002e12e89","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112097,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":22}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=22 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(22, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (22, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0098","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73355ae50f21fb5a17fbe27b939217801781c689f9462d211138487a6b0b4f5b","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112098,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":23}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=23 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(23, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (23, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0099","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7f14b1766c34de743187d9000324aff930d863d38b2d41d0b3211bf75e6a787","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112099,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":24}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=24 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(24, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (24, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r02-higher-order-derivative-0100","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89e34f2d971d7deb2d2243b3bf161442318be4d8dc44c68b7da704ee74333f90","family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112100,"semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":25}"},"fields":{"title":"A First Derivative May Exist Where the Second Does Not","stem":"Use nonzero scale=25 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).","answer":{"kind":"expression","value":"(25, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation":"The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (25, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)."},"skills":["A First Derivative May Exist Where the Second Does Not"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}