{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0001","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75836e7d0cab49a061468f920c523cad60ebd8b491e5d0a3232e01ec8dfa360e","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122001,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-12,\"derivatives\":[1,2,3],\"order\":2}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[1, 2, 3] at the fixed center=-12, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-12, 2, [1, 2, 3], 3, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-12, 2, [1, 2, 3], 3, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0002","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0b5f8e8607251f6a1424a33e7718ed99e0fc3600008dcc3d6d77191f4dc0d5b","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122002,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-11,\"derivatives\":[2,4,6,8],\"order\":3}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[2, 4, 6, 8] at the fixed center=-11, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-11, 3, [2, 4, 6, 8], 4, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-11, 3, [2, 4, 6, 8], 4, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0003","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9106d187cc3ea5ce7eb20d5ce6a2d39d6dd43cae8d0c7041aafb703baeda24c5","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122003,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-10,\"derivatives\":[3,6,9,12,15],\"order\":4}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[3, 6, 9, 12, 15] at the fixed center=-10, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-10, 4, [3, 6, 9, 12, 15], 5, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-10, 4, [3, 6, 9, 12, 15], 5, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0004","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0357c0834b56f998bdb03fc6ac5cb7aaa4a602c0c8d68758f4d9794257d1d30b","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122004,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-9,\"derivatives\":[4,8,12,16,20,24],\"order\":5}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[4, 8, 12, 16, 20, 24] at the fixed center=-9, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-9, 5, [4, 8, 12, 16, 20, 24], 6, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-9, 5, [4, 8, 12, 16, 20, 24], 6, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0005","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94151b0cdb13d8eb99e223dd11998431058800f967a8367d6778d65572410d8f","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122005,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-8,\"derivatives\":[5,10,15],\"order\":2}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[5, 10, 15] at the fixed center=-8, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-8, 2, [5, 10, 15], 3, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-8, 2, [5, 10, 15], 3, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0006","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eef205c1ff0d797fdb35d9837709fdd59ba7efdaad5a8b13b00a9ef758b84abc","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122006,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-7,\"derivatives\":[6,12,18,24],\"order\":3}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[6, 12, 18, 24] at the fixed center=-7, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-7, 3, [6, 12, 18, 24], 4, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-7, 3, [6, 12, 18, 24], 4, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0007","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea816db17af5d000ffefabaa44b3453ce63b9784da4c514ebdc4c9e155437d7a","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122007,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-6,\"derivatives\":[7,14,21,28,35],\"order\":4}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[7, 14, 21, 28, 35] at the fixed center=-6, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-6, 4, [7, 14, 21, 28, 35], 5, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-6, 4, [7, 14, 21, 28, 35], 5, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0008","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d488890ee4a3e3262b0c442af7b1df25215136579a4ea89e28dfb3bec23f6010","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122008,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-5,\"derivatives\":[8,16,24,32,40,48],\"order\":5}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[8, 16, 24, 32, 40, 48] at the fixed center=-5, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-5, 5, [8, 16, 24, 32, 40, 48], 6, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-5, 5, [8, 16, 24, 32, 40, 48], 6, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0009","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b57f874ed64a48e68b93b00040cfc77c60be87dd405d878cf3904ecce4cf4ea0","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122009,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-4,\"derivatives\":[9,18,27],\"order\":2}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[9, 18, 27] at the fixed center=-4, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-4, 2, [9, 18, 27], 3, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-4, 2, [9, 18, 27], 3, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0010","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcfc51478ddac2711303e50fcaffbb4029f54fb877e127b7a646ed4e56249e0a","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122010,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-3,\"derivatives\":[10,20,30,40],\"order\":3}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[10, 20, 30, 40] at the fixed center=-3, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-3, 3, [10, 20, 30, 40], 4, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-3, 3, [10, 20, 30, 40], 4, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0011","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"607ccfc7b0afd02ff1cbce04f00ae12c5b48a734cd6cab752871d7ee3b55b2cf","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122011,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-2,\"derivatives\":[11,22,33,44,55],\"order\":4}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[11, 22, 33, 44, 55] at the fixed center=-2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-2, 4, [11, 22, 33, 44, 55], 5, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-2, 4, [11, 22, 33, 44, 55], 5, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0012","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a12ed423b388d8b2e05de87675a188f9f589d496d468d76a389f7305f9eaade","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122012,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-1,\"derivatives\":[12,24,36,48,60,72],\"order\":5}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[12, 24, 36, 48, 60, 72] at the fixed center=-1, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(-1, 5, [12, 24, 36, 48, 60, 72], 6, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (-1, 5, [12, 24, 36, 48, 60, 72], 6, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0013","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5f49c9446bffc66cea69cf35557f541402ce4c981f00d9c982cd5f2a9eeb59d","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122013,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":0,\"derivatives\":[13,26,39],\"order\":2}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[13, 26, 39] at the fixed center=0, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(0, 2, [13, 26, 39], 3, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (0, 2, [13, 26, 39], 3, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0014","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2185404e2a4d60f511f0699200b023f25613f9b3a346c6b3f1ff13d0c0e55673","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122014,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":1,\"derivatives\":[14,28,42,56],\"order\":3}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[14, 28, 42, 56] at the fixed center=1, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(1, 3, [14, 28, 42, 56], 4, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (1, 3, [14, 28, 42, 56], 4, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0015","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"968b7d6be0ee722b37fc605f39601d3ecd8237104a8dadc5f97a3d4cf726aa35","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122015,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":2,\"derivatives\":[15,30,45,60,75],\"order\":4}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[15, 30, 45, 60, 75] at the fixed center=2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(2, 4, [15, 30, 45, 60, 75], 5, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (2, 4, [15, 30, 45, 60, 75], 5, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0016","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a48b53ebc92cbe612a637d440c1cfd2819bf60e4a07eccfa806d8c323d2327c8","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122016,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":3,\"derivatives\":[16,32,48,64,80,96],\"order\":5}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[16, 32, 48, 64, 80, 96] at the fixed center=3, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(3, 5, [16, 32, 48, 64, 80, 96], 6, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (3, 5, [16, 32, 48, 64, 80, 96], 6, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0017","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abecdf4b2a7d4c921758490fbd3865dc5460d5650480233d8e6864e15432ef83","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122017,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":4,\"derivatives\":[17,34,51],\"order\":2}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[17, 34, 51] at the fixed center=4, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(4, 2, [17, 34, 51], 3, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (4, 2, [17, 34, 51], 3, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0018","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de2ca7e47342df2ebdf41d7a6b433b9d47cba6289258af6bd9f86a6bf20760cd","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122018,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":5,\"derivatives\":[18,36,54,72],\"order\":3}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[18, 36, 54, 72] at the fixed center=5, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(5, 3, [18, 36, 54, 72], 4, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (5, 3, [18, 36, 54, 72], 4, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0019","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f615555a0cca3c4fd73f2acaaf81b33afdcdb322b65db6ac4be35828b9330efb","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122019,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":6,\"derivatives\":[19,38,57,76,95],\"order\":4}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[19, 38, 57, 76, 95] at the fixed center=6, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(6, 4, [19, 38, 57, 76, 95], 5, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (6, 4, [19, 38, 57, 76, 95], 5, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0020","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dd2c88cd1e5cc133188a89675cb5c514d2fee598b5c6d9a671407b36790c406","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122020,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":7,\"derivatives\":[20,40,60,80,100,120],\"order\":5}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[20, 40, 60, 80, 100, 120] at the fixed center=7, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(7, 5, [20, 40, 60, 80, 100, 120], 6, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (7, 5, [20, 40, 60, 80, 100, 120], 6, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0021","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"954475f7d73bb61155c95032ac7c9b15eb273ac078d17952c50639317503b8d5","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122021,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":8,\"derivatives\":[21,42,63],\"order\":2}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[21, 42, 63] at the fixed center=8, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(8, 2, [21, 42, 63], 3, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (8, 2, [21, 42, 63], 3, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0022","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79b9183869a3c5d557156bde4512b1a596cbce59c2ae95d6c2587670169c80a6","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122022,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":9,\"derivatives\":[22,44,66,88],\"order\":3}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[22, 44, 66, 88] at the fixed center=9, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(9, 3, [22, 44, 66, 88], 4, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (9, 3, [22, 44, 66, 88], 4, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0023","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82ef582e2975a9d048d0bf62f217338b79487370f4499b02de4908b495acafb5","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122023,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":10,\"derivatives\":[23,46,69,92,115],\"order\":4}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[23, 46, 69, 92, 115] at the fixed center=10, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(10, 4, [23, 46, 69, 92, 115], 5, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (10, 4, [23, 46, 69, 92, 115], 5, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0024","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2255e9318e95e6fa9365495418c3efd69bfbc665bca8a25048e48c85baa2671e","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122024,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":11,\"derivatives\":[24,48,72,96,120,144],\"order\":5}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[24, 48, 72, 96, 120, 144] at the fixed center=11, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(11, 5, [24, 48, 72, 96, 120, 144], 6, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (11, 5, [24, 48, 72, 96, 120, 144], 6, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0025","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce6bfcc09cd60003ced3379dcdc43e81b028ab90b180625616a828001c32fe50","family_id":"qf-mod43-taylor-jet-qualification","seed":202611122025,"semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":12,\"derivatives\":[25,50,75],\"order\":2}"},"fields":{"title":"A Taylor Jet Includes Every Derivative Order from Zero through n","stem":"Use the derivative values=[25, 50, 75] at the fixed center=12, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).","answer":{"kind":"expression","value":"(12, 2, [25, 50, 75], 3, Yes, complete_zero_through_n_derivative_jet)","explanation":"Check exact jet arity n+1 and its declared center. At n=0 the polynomial is the constant f(a). Completeness does not assert a remainder bound, infinite-series convergence or analyticity. The exact result, retaining the requested order, is (12, 2, [25, 50, 75], 3, Yes, complete_zero_through_n_derivative_jet)."},"skills":["A Taylor Jet Includes Every Derivative Order from Zero through n"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0026","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e171ada7daaef58e87f9a1333556cab31662e193ac65e9eea4792ffe02e3045b","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122026,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-12,\"derivatives\":[1,2,3],\"order\":2}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[1, 2, 3] at the fixed center=-12, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-12, 2, [1, 2, 3/2], [[0, 1], [1, 2], [2, 3/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-12, 2, [1, 2, 3/2], [[0, 1], [1, 2], [2, 3/2]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0027","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"300aa8fc4947e9a3cef5a06bfe8d28d5a5932c055af49296869d15ad50e725e8","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122027,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-11,\"derivatives\":[2,4,6,8],\"order\":3}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[2, 4, 6, 8] at the fixed center=-11, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-11, 3, [2, 4, 3, 4/3], [[0, 2], [1, 4], [2, 3], [3, 4/3]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-11, 3, [2, 4, 3, 4/3], [[0, 2], [1, 4], [2, 3], [3, 4/3]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0028","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26f21cc48c618a92f1f3114963956416d8e1953d2d7b55fe6f84d368022bcd0d","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122028,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-10,\"derivatives\":[3,6,9,12,15],\"order\":4}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[3, 6, 9, 12, 15] at the fixed center=-10, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-10, 4, [3, 6, 9/2, 2, 5/8], [[0, 3], [1, 6], [2, 9/2], [3, 2], [4, 5/8]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-10, 4, [3, 6, 9/2, 2, 5/8], [[0, 3], [1, 6], [2, 9/2], [3, 2], [4, 5/8]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0029","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41ebc72223a1f8fc99c19becd1328c7720a6e5e117399780e5c71b2e5916c307","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122029,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-9,\"derivatives\":[4,8,12,16,20,24],\"order\":5}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[4, 8, 12, 16, 20, 24] at the fixed center=-9, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-9, 5, [4, 8, 6, 8/3, 5/6, 1/5], [[0, 4], [1, 8], [2, 6], [3, 8/3], [4, 5/6], [5, 1/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-9, 5, [4, 8, 6, 8/3, 5/6, 1/5], [[0, 4], [1, 8], [2, 6], [3, 8/3], [4, 5/6], [5, 1/5]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0030","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83671c38c4191176d2d3fe58eb1880b4296abc17ff2809a131eea46c7fc1c3fd","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122030,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-8,\"derivatives\":[5,10,15],\"order\":2}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[5, 10, 15] at the fixed center=-8, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0031","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"234be15f4721d2bf662cadf08c697d93f351747e91b0ba2cac8d6e0a0e9cef55","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122031,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-7,\"derivatives\":[6,12,18,24],\"order\":3}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[6, 12, 18, 24] at the fixed center=-7, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-7, 3, [6, 12, 9, 4], [[0, 6], [1, 12], [2, 9], [3, 4]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-7, 3, [6, 12, 9, 4], [[0, 6], [1, 12], [2, 9], [3, 4]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0032","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42f09ff8de6a34a239e74c31305c44c3ace2802aa5b7d8df5ea51e915d90ec38","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122032,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-6,\"derivatives\":[7,14,21,28,35],\"order\":4}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[7, 14, 21, 28, 35] at the fixed center=-6, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-6, 4, [7, 14, 21/2, 14/3, 35/24], [[0, 7], [1, 14], [2, 21/2], [3, 14/3], [4, 35/24]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-6, 4, [7, 14, 21/2, 14/3, 35/24], [[0, 7], [1, 14], [2, 21/2], [3, 14/3], [4, 35/24]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0033","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f32e59c78165cf896fadbf405c5046c531101b3026322a19328a4ef160982b50","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122033,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-5,\"derivatives\":[8,16,24,32,40,48],\"order\":5}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[8, 16, 24, 32, 40, 48] at the fixed center=-5, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-5, 5, [8, 16, 12, 16/3, 5/3, 2/5], [[0, 8], [1, 16], [2, 12], [3, 16/3], [4, 5/3], [5, 2/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-5, 5, [8, 16, 12, 16/3, 5/3, 2/5], [[0, 8], [1, 16], [2, 12], [3, 16/3], [4, 5/3], [5, 2/5]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0034","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f280a2fcdf69e0645f969e11d47e254f9878fc19da15d84e00615a0426e9cc7","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122034,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-4,\"derivatives\":[9,18,27],\"order\":2}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[9, 18, 27] at the fixed center=-4, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-4, 2, [9, 18, 27/2], [[0, 9], [1, 18], [2, 27/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-4, 2, [9, 18, 27/2], [[0, 9], [1, 18], [2, 27/2]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0035","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"002901a7a215cdb36e996267d3510040c05090fc1222c987e64514e3cd1b7b22","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122035,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-3,\"derivatives\":[10,20,30,40],\"order\":3}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[10, 20, 30, 40] at the fixed center=-3, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-3, 3, [10, 20, 15, 20/3], [[0, 10], [1, 20], [2, 15], [3, 20/3]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-3, 3, [10, 20, 15, 20/3], [[0, 10], [1, 20], [2, 15], [3, 20/3]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0036","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16f0a43513d682f41b36246293333d9f746b718b5e4117168b487b2c332f758b","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122036,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-2,\"derivatives\":[11,22,33,44,55],\"order\":4}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[11, 22, 33, 44, 55] at the fixed center=-2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-2, 4, [11, 22, 33/2, 22/3, 55/24], [[0, 11], [1, 22], [2, 33/2], [3, 22/3], [4, 55/24]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-2, 4, [11, 22, 33/2, 22/3, 55/24], [[0, 11], [1, 22], [2, 33/2], [3, 22/3], [4, 55/24]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0037","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"401ad3eba993d95c8040f16d29f24a3868368357dae0c9e905fb0dfe2c41dac7","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122037,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-1,\"derivatives\":[12,24,36,48,60,72],\"order\":5}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[12, 24, 36, 48, 60, 72] at the fixed center=-1, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0038","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b6f8238d7c3b5b0621c0a039e6c6008930602fc01a4207ae39cfab4bfb9b6dc","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122038,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":0,\"derivatives\":[13,26,39],\"order\":2}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[13, 26, 39] at the fixed center=0, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(0, 2, [13, 26, 39/2], [[0, 13], [1, 26], [2, 39/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (0, 2, [13, 26, 39/2], [[0, 13], [1, 26], [2, 39/2]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0039","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9383524dff77cf605c45d17aadd415cb6c03e046c8c8c989a12966686e34ee6e","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122039,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":1,\"derivatives\":[14,28,42,56],\"order\":3}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[14, 28, 42, 56] at the fixed center=1, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(1, 3, [14, 28, 21, 28/3], [[0, 14], [1, 28], [2, 21], [3, 28/3]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (1, 3, [14, 28, 21, 28/3], [[0, 14], [1, 28], [2, 21], [3, 28/3]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0040","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f7e4bb47abd8e84efde4af2a8438f60233047e818f012bd7d01fcdf5495d5d5","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122040,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":2,\"derivatives\":[15,30,45,60,75],\"order\":4}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[15, 30, 45, 60, 75] at the fixed center=2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0041","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69af7e4dc6f96814e7daac103c1f0710e06aad86dc4fbc270d51a33f66a12033","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122041,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":3,\"derivatives\":[16,32,48,64,80,96],\"order\":5}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[16, 32, 48, 64, 80, 96] at the fixed center=3, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(3, 5, [16, 32, 24, 32/3, 10/3, 4/5], [[0, 16], [1, 32], [2, 24], [3, 32/3], [4, 10/3], [5, 4/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (3, 5, [16, 32, 24, 32/3, 10/3, 4/5], [[0, 16], [1, 32], [2, 24], [3, 32/3], [4, 10/3], [5, 4/5]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0042","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12b9a98f855879524ec3e4cc1899451277da69a7a9be7d6714c32b376b815329","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122042,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":4,\"derivatives\":[17,34,51],\"order\":2}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[17, 34, 51] at the fixed center=4, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(4, 2, [17, 34, 51/2], [[0, 17], [1, 34], [2, 51/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (4, 2, [17, 34, 51/2], [[0, 17], [1, 34], [2, 51/2]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0043","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8f2f67cc10ef07b7d27512be10d3a77a0c1190354002062d755972af39b3dfd","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122043,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":5,\"derivatives\":[18,36,54,72],\"order\":3}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[18, 36, 54, 72] at the fixed center=5, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(5, 3, [18, 36, 27, 12], [[0, 18], [1, 36], [2, 27], [3, 12]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (5, 3, [18, 36, 27, 12], [[0, 18], [1, 36], [2, 27], [3, 12]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0044","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2967f19662a74bec75febd9f18c70dc9f5decbb15733d6cae75e0fb418713e41","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122044,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":6,\"derivatives\":[19,38,57,76,95],\"order\":4}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[19, 38, 57, 76, 95] at the fixed center=6, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(6, 4, [19, 38, 57/2, 38/3, 95/24], [[0, 19], [1, 38], [2, 57/2], [3, 38/3], [4, 95/24]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (6, 4, [19, 38, 57/2, 38/3, 95/24], [[0, 19], [1, 38], [2, 57/2], [3, 38/3], [4, 95/24]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0045","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"930e63ac00db550846ddcc2e93b1e7015165dedd985a791e40b020a6f9c26c5f","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122045,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":7,\"derivatives\":[20,40,60,80,100,120],\"order\":5}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[20, 40, 60, 80, 100, 120] at the fixed center=7, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(7, 5, [20, 40, 30, 40/3, 25/6, 1], [[0, 20], [1, 40], [2, 30], [3, 40/3], [4, 25/6], [5, 1]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (7, 5, [20, 40, 30, 40/3, 25/6, 1], [[0, 20], [1, 40], [2, 30], [3, 40/3], [4, 25/6], [5, 1]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0046","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37ec165e0596671931e9cd7b022bd60831a38b845d7e37689704a713b6bb859a","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122046,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":8,\"derivatives\":[21,42,63],\"order\":2}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[21, 42, 63] at the fixed center=8, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(8, 2, [21, 42, 63/2], [[0, 21], [1, 42], [2, 63/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (8, 2, [21, 42, 63/2], [[0, 21], [1, 42], [2, 63/2]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0047","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd09aef4c9db0643d7318fcbb5ccaab3aa24304c4f7e13674aa492f84f0ac54e","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122047,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":9,\"derivatives\":[22,44,66,88],\"order\":3}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[22, 44, 66, 88] at the fixed center=9, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(9, 3, [22, 44, 33, 44/3], [[0, 22], [1, 44], [2, 33], [3, 44/3]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (9, 3, [22, 44, 33, 44/3], [[0, 22], [1, 44], [2, 33], [3, 44/3]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0048","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca73367594922f79e611d29a2b13fa19e3e4f13419f85e6c042eee2b43de5fca","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122048,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":10,\"derivatives\":[23,46,69,92,115],\"order\":4}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[23, 46, 69, 92, 115] at the fixed center=10, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(10, 4, [23, 46, 69/2, 46/3, 115/24], [[0, 23], [1, 46], [2, 69/2], [3, 46/3], [4, 115/24]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (10, 4, [23, 46, 69/2, 46/3, 115/24], [[0, 23], [1, 46], [2, 69/2], [3, 46/3], [4, 115/24]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0049","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"527a19cdcca8a94985653c66462b18bdaf611e0b14b37271bdaa38516a7c875b","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122049,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":11,\"derivatives\":[24,48,72,96,120,144],\"order\":5}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[24, 48, 72, 96, 120, 144] at the fixed center=11, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(11, 5, [24, 48, 36, 16, 5, 6/5], [[0, 24], [1, 48], [2, 36], [3, 16], [4, 5], [5, 6/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (11, 5, [24, 48, 36, 16, 5, 6/5], [[0, 24], [1, 48], [2, 36], [3, 16], [4, 5], [5, 6/5]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0050","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"149860c17a4c08b73abfada69c139a7c0c76a86c3435f465c5df74383f510e4d","family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122050,"semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":12,\"derivatives\":[25,50,75],\"order\":2}"},"fields":{"title":"Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial","stem":"Use the derivative values=[25, 50, 75] at the fixed center=12, through declared order=2≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).","answer":{"kind":"expression","value":"(12, 2, [25, 50, 75/2], [[0, 25], [1, 50], [2, 75/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation":"Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (12, 2, [25, 50, 75/2], [[0, 25], [1, 50], [2, 75/2]], coefficient_k_equals_derivative_k_over_k_factorial)."},"skills":["Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0051","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f1a1caf1be170e27d1f058612e68339325b2fad176ea7eae06c4caaf05accb1","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122051,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-12,\"point\":-14,\"shifted_coefficients\":[1,2,3]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[1, 2, 3] of the explicit polynomial sum c_k*(x−a)^k, center a=-12, evaluation point=-14. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], [1, 2, 6], [1, 2, 3], -14, 9, 9, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([1, 2, 3], [1, 2, 6], [1, 2, 3], -14, 9, 9, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0052","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb6f775406de6afccb42a723fef254a47bb54c53e631a481107cd5e3b6ce29bb","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122052,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-11,\"point\":-12,\"shifted_coefficients\":[2,4,6,8]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[2, 4, 6, 8] of the explicit polynomial sum c_k*(x−a)^k, center a=-11, evaluation point=-12. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0053","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d113ea452b49ce61d07298f7b289dcc378bb41e61a8b996a6558f725c75c5f62","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122053,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-10,\"point\":-10,\"shifted_coefficients\":[3,6,9,12,15]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[3, 6, 9, 12, 15] of the explicit polynomial sum c_k*(x−a)^k, center a=-10, evaluation point=-10. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([3, 6, 9, 12, 15], [3, 6, 18, 72, 360], [3, 6, 9, 12, 15], -10, 3, 3, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([3, 6, 9, 12, 15], [3, 6, 18, 72, 360], [3, 6, 9, 12, 15], -10, 3, 3, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0054","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5455ca1324ca280eae43fd7c8ed8d49c0259c1cf4843fb8d808e071ab9702253","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122054,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-9,\"point\":-8,\"shifted_coefficients\":[4,8,12,16,20,24]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[4, 8, 12, 16, 20, 24] of the explicit polynomial sum c_k*(x−a)^k, center a=-9, evaluation point=-8. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([4, 8, 12, 16, 20, 24], [4, 8, 24, 96, 480, 2880], [4, 8, 12, 16, 20, 24], -8, 84, 84, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([4, 8, 12, 16, 20, 24], [4, 8, 24, 96, 480, 2880], [4, 8, 12, 16, 20, 24], -8, 84, 84, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0055","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1e84a0b8fe89f841c7c18c26e706bee6e2830b874cf50e047da1f0231803dc6","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122055,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-8,\"point\":-6,\"shifted_coefficients\":[5,10,15]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[5, 10, 15] of the explicit polynomial sum c_k*(x−a)^k, center a=-8, evaluation point=-6. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([5, 10, 15], [5, 10, 30], [5, 10, 15], -6, 85, 85, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([5, 10, 15], [5, 10, 30], [5, 10, 15], -6, 85, 85, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0056","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00bb0ed1abfa2201b1e1757198aed46e68eeb64176dc89c17c6b19e1a4220d6b","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122056,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-7,\"point\":-9,\"shifted_coefficients\":[6,12,18,24]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[6, 12, 18, 24] of the explicit polynomial sum c_k*(x−a)^k, center a=-7, evaluation point=-9. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([6, 12, 18, 24], [6, 12, 36, 144], [6, 12, 18, 24], -9, -138, -138, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([6, 12, 18, 24], [6, 12, 36, 144], [6, 12, 18, 24], -9, -138, -138, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0057","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd652bb9ec47a3ff44ef0bd7c5ded681e7637e12d591beec63b2c9022154e1a8","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122057,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-6,\"point\":-7,\"shifted_coefficients\":[7,14,21,28,35]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[7, 14, 21, 28, 35] of the explicit polynomial sum c_k*(x−a)^k, center a=-6, evaluation point=-7. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([7, 14, 21, 28, 35], [7, 14, 42, 168, 840], [7, 14, 21, 28, 35], -7, 21, 21, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([7, 14, 21, 28, 35], [7, 14, 42, 168, 840], [7, 14, 21, 28, 35], -7, 21, 21, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0058","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdc1e9d9ca5bfc9196d7b6d6ec491d4c65f67ef0a0d2235432f81450388f36c1","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122058,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-5,\"point\":-5,\"shifted_coefficients\":[8,16,24,32,40,48]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[8, 16, 24, 32, 40, 48] of the explicit polynomial sum c_k*(x−a)^k, center a=-5, evaluation point=-5. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([8, 16, 24, 32, 40, 48], [8, 16, 48, 192, 960, 5760], [8, 16, 24, 32, 40, 48], -5, 8, 8, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([8, 16, 24, 32, 40, 48], [8, 16, 48, 192, 960, 5760], [8, 16, 24, 32, 40, 48], -5, 8, 8, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0059","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bd421176f6bef4c43db3769c479a5085c7e591a91f241ce1223e9752261d389","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122059,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-4,\"point\":-3,\"shifted_coefficients\":[9,18,27]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[9, 18, 27] of the explicit polynomial sum c_k*(x−a)^k, center a=-4, evaluation point=-3. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([9, 18, 27], [9, 18, 54], [9, 18, 27], -3, 54, 54, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([9, 18, 27], [9, 18, 54], [9, 18, 27], -3, 54, 54, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0060","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee0a44271f82043cad5e5792e415147eb9a756436e2cd1b302c26a87228148ca","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122060,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-3,\"point\":-1,\"shifted_coefficients\":[10,20,30,40]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[10, 20, 30, 40] of the explicit polynomial sum c_k*(x−a)^k, center a=-3, evaluation point=-1. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0061","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"225895b9cf54781f1945823fbb5cb88bcf5469d54682cc01296e3f55e0d307be","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122061,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-2,\"point\":-4,\"shifted_coefficients\":[11,22,33,44,55]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[11, 22, 33, 44, 55] of the explicit polynomial sum c_k*(x−a)^k, center a=-2, evaluation point=-4. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([11, 22, 33, 44, 55], [11, 22, 66, 264, 1320], [11, 22, 33, 44, 55], -4, 627, 627, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([11, 22, 33, 44, 55], [11, 22, 66, 264, 1320], [11, 22, 33, 44, 55], -4, 627, 627, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0062","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90f5a111251d895a4a475bd1503fbd0c40e88f011d297bb14c17dec08222bc5b","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122062,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-1,\"point\":-2,\"shifted_coefficients\":[12,24,36,48,60,72]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[12, 24, 36, 48, 60, 72] of the explicit polynomial sum c_k*(x−a)^k, center a=-1, evaluation point=-2. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([12, 24, 36, 48, 60, 72], [12, 24, 72, 288, 1440, 8640], [12, 24, 36, 48, 60, 72], -2, -36, -36, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([12, 24, 36, 48, 60, 72], [12, 24, 72, 288, 1440, 8640], [12, 24, 36, 48, 60, 72], -2, -36, -36, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0063","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59eb43bbcea0256a16b938f931acda72258fcb6b44ec1f2689bede308bec24eb","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122063,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":0,\"point\":0,\"shifted_coefficients\":[13,26,39]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[13, 26, 39] of the explicit polynomial sum c_k*(x−a)^k, center a=0, evaluation point=0. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([13, 26, 39], [13, 26, 78], [13, 26, 39], 0, 13, 13, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([13, 26, 39], [13, 26, 78], [13, 26, 39], 0, 13, 13, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0064","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94df77c9f7aebbb48f7453881db998aa643a3fd453147188983c731553813889","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122064,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":1,\"point\":2,\"shifted_coefficients\":[14,28,42,56]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[14, 28, 42, 56] of the explicit polynomial sum c_k*(x−a)^k, center a=1, evaluation point=2. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0065","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79252fa37ed2626610846c81dc5293394bad172e3486fe43cc2a6ac66ce0654b","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122065,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":2,\"point\":4,\"shifted_coefficients\":[15,30,45,60,75]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[15, 30, 45, 60, 75] of the explicit polynomial sum c_k*(x−a)^k, center a=2, evaluation point=4. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([15, 30, 45, 60, 75], [15, 30, 90, 360, 1800], [15, 30, 45, 60, 75], 4, 1935, 1935, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([15, 30, 45, 60, 75], [15, 30, 90, 360, 1800], [15, 30, 45, 60, 75], 4, 1935, 1935, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0066","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36935940cec161a7f52815203b199e23863da0c7c99c12e6474b696c164a4c3c","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122066,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":3,\"point\":1,\"shifted_coefficients\":[16,32,48,64,80,96]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[16, 32, 48, 64, 80, 96] of the explicit polynomial sum c_k*(x−a)^k, center a=3, evaluation point=1. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0067","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84daf705f82609e0630a3d8b5fb71008410111d3299fb4cfa7e262b6a212acbf","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122067,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":4,\"point\":3,\"shifted_coefficients\":[17,34,51]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[17, 34, 51] of the explicit polynomial sum c_k*(x−a)^k, center a=4, evaluation point=3. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([17, 34, 51], [17, 34, 102], [17, 34, 51], 3, 34, 34, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([17, 34, 51], [17, 34, 102], [17, 34, 51], 3, 34, 34, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0068","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b2436437acb6cba678307e1438fae8f81f88d2269451faee1deaa128bf2c9c","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122068,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":5,\"point\":5,\"shifted_coefficients\":[18,36,54,72]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[18, 36, 54, 72] of the explicit polynomial sum c_k*(x−a)^k, center a=5, evaluation point=5. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([18, 36, 54, 72], [18, 36, 108, 432], [18, 36, 54, 72], 5, 18, 18, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([18, 36, 54, 72], [18, 36, 108, 432], [18, 36, 54, 72], 5, 18, 18, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0069","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"394d2399262255d849f90dc51642348823b422d9ce1e2a54251dad82acd7ffb8","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122069,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":6,\"point\":7,\"shifted_coefficients\":[19,38,57,76,95]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[19, 38, 57, 76, 95] of the explicit polynomial sum c_k*(x−a)^k, center a=6, evaluation point=7. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([19, 38, 57, 76, 95], [19, 38, 114, 456, 2280], [19, 38, 57, 76, 95], 7, 285, 285, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([19, 38, 57, 76, 95], [19, 38, 114, 456, 2280], [19, 38, 57, 76, 95], 7, 285, 285, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0070","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d37e7cc91b18be63d7292d57e9ebffe692aff37e2a10c7b87f3e4a6d8f11fe6","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122070,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":7,\"point\":9,\"shifted_coefficients\":[20,40,60,80,100,120]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[20, 40, 60, 80, 100, 120] of the explicit polynomial sum c_k*(x−a)^k, center a=7, evaluation point=9. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([20, 40, 60, 80, 100, 120], [20, 40, 120, 480, 2400, 14400], [20, 40, 60, 80, 100, 120], 9, 6420, 6420, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([20, 40, 60, 80, 100, 120], [20, 40, 120, 480, 2400, 14400], [20, 40, 60, 80, 100, 120], 9, 6420, 6420, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0071","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88ffd01ad2a598ba4f3927cb7972e09eba9cd4c121b5465429d42a1719139307","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122071,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":8,\"point\":6,\"shifted_coefficients\":[21,42,63]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[21, 42, 63] of the explicit polynomial sum c_k*(x−a)^k, center a=8, evaluation point=6. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([21, 42, 63], [21, 42, 126], [21, 42, 63], 6, 189, 189, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([21, 42, 63], [21, 42, 126], [21, 42, 63], 6, 189, 189, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0072","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1d86d31f1039f25296c7603ddcd09e7837605910c8a68c47971c279d8a334d2","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122072,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":9,\"point\":8,\"shifted_coefficients\":[22,44,66,88]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[22, 44, 66, 88] of the explicit polynomial sum c_k*(x−a)^k, center a=9, evaluation point=8. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([22, 44, 66, 88], [22, 44, 132, 528], [22, 44, 66, 88], 8, -44, -44, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([22, 44, 66, 88], [22, 44, 132, 528], [22, 44, 66, 88], 8, -44, -44, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0073","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ce1b0d3c8721a5695ece8752d7c6e37ca526b0bdf56c303b3dec830f7bee5f5","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122073,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":10,\"point\":10,\"shifted_coefficients\":[23,46,69,92,115]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[23, 46, 69, 92, 115] of the explicit polynomial sum c_k*(x−a)^k, center a=10, evaluation point=10. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([23, 46, 69, 92, 115], [23, 46, 138, 552, 2760], [23, 46, 69, 92, 115], 10, 23, 23, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([23, 46, 69, 92, 115], [23, 46, 138, 552, 2760], [23, 46, 69, 92, 115], 10, 23, 23, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0074","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cd986b0a54b71a857ffba4dc00972b40e8bac8972db8fc4ae86b28456bdc8c2","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122074,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":11,\"point\":12,\"shifted_coefficients\":[24,48,72,96,120,144]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[24, 48, 72, 96, 120, 144] of the explicit polynomial sum c_k*(x−a)^k, center a=11, evaluation point=12. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([24, 48, 72, 96, 120, 144], [24, 48, 144, 576, 2880, 17280], [24, 48, 72, 96, 120, 144], 12, 504, 504, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([24, 48, 72, 96, 120, 144], [24, 48, 144, 576, 2880, 17280], [24, 48, 72, 96, 120, 144], 12, 504, 504, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0075","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95335b628806126b464e8bf01339c61d3e7867a525f1f611d274b2cb9d77a68b","family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122075,"semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":12,\"point\":14,\"shifted_coefficients\":[25,50,75]}"},"fields":{"title":"A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet","stem":"Use shifted_coefficients=[25, 50, 75] of the explicit polynomial sum c_k*(x−a)^k, center a=12, evaluation point=14. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).","answer":{"kind":"expression","value":"([25, 50, 75], [25, 50, 150], [25, 50, 75], 14, 425, 425, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation":"For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([25, 50, 75], [25, 50, 150], [25, 50, 75], 14, 425, 425, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."},"skills":["A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0076","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62212f8123df9ce0e1be7e9c8fcb7c5cc5aee9b6a9717c2ebd48a3e4627e6705","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122076,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-11,\"center\":-12,\"derivatives\":[1,2],\"mode\":\"missing_derivative\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_derivative, supplied derivatives=[1, 2], center=-12, order=2, alternate_center=-11. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_derivative, 2, [1, 2], -12, -11, No, all_derivatives_zero_through_n_are_required)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [1, 2], -12, -11, No, all_derivatives_zero_through_n_are_required)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0077","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38d52ecd14a0a5e7dcf86153bc3f802c40641c45566431cdd0ce1f765093188d","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122077,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-10,\"center\":-11,\"derivatives\":[2,4,6],\"mode\":\"wrong_center\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=wrong_center, supplied derivatives=[2, 4, 6], center=-11, order=2, alternate_center=-10. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(wrong_center, 2, [2, 4, 6], -11, -10, No, a_jet_is_bound_to_its_declared_center)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [2, 4, 6], -11, -10, No, a_jet_is_bound_to_its_declared_center)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0078","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef3e37c3bfd2192ee4a232383515e85dd18a4c486034a5ee5a7ef53914dc5bf1","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122078,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-9,\"center\":-10,\"derivatives\":[3,6,9],\"mode\":\"missing_factorial\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_factorial, supplied derivatives=[3, 6, 9], center=-10, order=2, alternate_center=-9. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_factorial, 2, [3, 6, 9], -10, -9, No, coefficients_require_division_by_k_factorial)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [3, 6, 9], -10, -9, No, coefficients_require_division_by_k_factorial)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0079","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96280d6c287142c21ff60b8824c058f3ccafad44677815b01beddf73c1736488","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122079,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-8,\"center\":-9,\"derivatives\":[4,8,12],\"mode\":\"remainder_claim\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=remainder_claim, supplied derivatives=[4, 8, 12], center=-9, order=2, alternate_center=-8. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(remainder_claim, 2, [4, 8, 12], -9, -8, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [4, 8, 12], -9, -8, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0080","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fecc59c141ae2b883662c3a33f82132a6f3e7809c005ba84a5849b20fab38d9b","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122080,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-7,\"center\":-8,\"derivatives\":[5,10],\"mode\":\"missing_derivative\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_derivative, supplied derivatives=[5, 10], center=-8, order=2, alternate_center=-7. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_derivative, 2, [5, 10], -8, -7, No, all_derivatives_zero_through_n_are_required)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [5, 10], -8, -7, No, all_derivatives_zero_through_n_are_required)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0081","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10a2da4ffc435afa2a84e2a350219d6d34a56487652e6efcd26246ec06d2f866","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122081,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-6,\"center\":-7,\"derivatives\":[6,12,18],\"mode\":\"wrong_center\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=wrong_center, supplied derivatives=[6, 12, 18], center=-7, order=2, alternate_center=-6. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(wrong_center, 2, [6, 12, 18], -7, -6, No, a_jet_is_bound_to_its_declared_center)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [6, 12, 18], -7, -6, No, a_jet_is_bound_to_its_declared_center)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0082","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad3852c3fb2c1caddce9faa7a4dc5ff71da03786865b96f1450848227c8fc3fd","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122082,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-5,\"center\":-6,\"derivatives\":[7,14,21],\"mode\":\"missing_factorial\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_factorial, supplied derivatives=[7, 14, 21], center=-6, order=2, alternate_center=-5. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_factorial, 2, [7, 14, 21], -6, -5, No, coefficients_require_division_by_k_factorial)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [7, 14, 21], -6, -5, No, coefficients_require_division_by_k_factorial)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0083","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96ed7b117627c9f9d84e91f4645643cdfa1ea8315e192c06c6ec99807d622ffa","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122083,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-4,\"center\":-5,\"derivatives\":[8,16,24],\"mode\":\"remainder_claim\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=remainder_claim, supplied derivatives=[8, 16, 24], center=-5, order=2, alternate_center=-4. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(remainder_claim, 2, [8, 16, 24], -5, -4, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [8, 16, 24], -5, -4, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0084","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"690a2f8ae8989c9bd944ffdcf166698ce6386007de94f63d9eceb5d7f7e0f217","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122084,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-3,\"center\":-4,\"derivatives\":[9,18],\"mode\":\"missing_derivative\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_derivative, supplied derivatives=[9, 18], center=-4, order=2, alternate_center=-3. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_derivative, 2, [9, 18], -4, -3, No, all_derivatives_zero_through_n_are_required)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [9, 18], -4, -3, No, all_derivatives_zero_through_n_are_required)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0085","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ca07e3ee244c1184323d0f05efb15dfac2a80b8ad8c84592530c558f6a61864","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122085,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-2,\"center\":-3,\"derivatives\":[10,20,30],\"mode\":\"wrong_center\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=wrong_center, supplied derivatives=[10, 20, 30], center=-3, order=2, alternate_center=-2. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(wrong_center, 2, [10, 20, 30], -3, -2, No, a_jet_is_bound_to_its_declared_center)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [10, 20, 30], -3, -2, No, a_jet_is_bound_to_its_declared_center)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0086","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"971b1ce5030b690dcbe3dc4a69631b962df85e801ce440d7d40db8c2fb9da205","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122086,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-1,\"center\":-2,\"derivatives\":[11,22,33],\"mode\":\"missing_factorial\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_factorial, supplied derivatives=[11, 22, 33], center=-2, order=2, alternate_center=-1. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_factorial, 2, [11, 22, 33], -2, -1, No, coefficients_require_division_by_k_factorial)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [11, 22, 33], -2, -1, No, coefficients_require_division_by_k_factorial)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0087","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d3d22afcb17ca5eb6490e5b8efe591aec2d3dfcf94d1edb401f5a974e808f6","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122087,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":0,\"center\":-1,\"derivatives\":[12,24,36],\"mode\":\"remainder_claim\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=remainder_claim, supplied derivatives=[12, 24, 36], center=-1, order=2, alternate_center=0. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(remainder_claim, 2, [12, 24, 36], -1, 0, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [12, 24, 36], -1, 0, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0088","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"387781461f25ec2a47d8bb470378715d7235ae0052b2396cf799a0a88e76a034","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122088,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":1,\"center\":0,\"derivatives\":[13,26],\"mode\":\"missing_derivative\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_derivative, supplied derivatives=[13, 26], center=0, order=2, alternate_center=1. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_derivative, 2, [13, 26], 0, 1, No, all_derivatives_zero_through_n_are_required)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [13, 26], 0, 1, No, all_derivatives_zero_through_n_are_required)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0089","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"889b0f8b8998d3d148c5001d029d405f1db8b6596ecdd5dad6f3039df8d890ea","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122089,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":2,\"center\":1,\"derivatives\":[14,28,42],\"mode\":\"wrong_center\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=wrong_center, supplied derivatives=[14, 28, 42], center=1, order=2, alternate_center=2. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(wrong_center, 2, [14, 28, 42], 1, 2, No, a_jet_is_bound_to_its_declared_center)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [14, 28, 42], 1, 2, No, a_jet_is_bound_to_its_declared_center)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0090","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6512f1f9ba1c5c4cf60bc2abc4f0e97eed82df2c3508719efb3ddf0702eb148","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122090,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":3,\"center\":2,\"derivatives\":[15,30,45],\"mode\":\"missing_factorial\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_factorial, supplied derivatives=[15, 30, 45], center=2, order=2, alternate_center=3. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_factorial, 2, [15, 30, 45], 2, 3, No, coefficients_require_division_by_k_factorial)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [15, 30, 45], 2, 3, No, coefficients_require_division_by_k_factorial)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0091","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eed36193267ee8c03baaba12bc0fb9ea0e83bdd07a2eb820a8e7e1212c7585b","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122091,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":4,\"center\":3,\"derivatives\":[16,32,48],\"mode\":\"remainder_claim\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=remainder_claim, supplied derivatives=[16, 32, 48], center=3, order=2, alternate_center=4. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(remainder_claim, 2, [16, 32, 48], 3, 4, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [16, 32, 48], 3, 4, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0092","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93c1980103eccd3ede98a31c318bc79afdebfceed9293a6a1b15b6b360ed1e66","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122092,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":5,\"center\":4,\"derivatives\":[17,34],\"mode\":\"missing_derivative\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_derivative, supplied derivatives=[17, 34], center=4, order=2, alternate_center=5. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_derivative, 2, [17, 34], 4, 5, No, all_derivatives_zero_through_n_are_required)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [17, 34], 4, 5, No, all_derivatives_zero_through_n_are_required)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0093","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3c374e98aee83defdb44a1c45fe31a0e9f4eb2fe6100f84a67cd7cb80bb390c","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122093,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":6,\"center\":5,\"derivatives\":[18,36,54],\"mode\":\"wrong_center\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=wrong_center, supplied derivatives=[18, 36, 54], center=5, order=2, alternate_center=6. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(wrong_center, 2, [18, 36, 54], 5, 6, No, a_jet_is_bound_to_its_declared_center)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [18, 36, 54], 5, 6, No, a_jet_is_bound_to_its_declared_center)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0094","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe7808a3365f2a80785258c373c42b88a1af371d2c3fb8dbd14dd72c68f3afcd","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122094,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":7,\"center\":6,\"derivatives\":[19,38,57],\"mode\":\"missing_factorial\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_factorial, supplied derivatives=[19, 38, 57], center=6, order=2, alternate_center=7. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_factorial, 2, [19, 38, 57], 6, 7, No, coefficients_require_division_by_k_factorial)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [19, 38, 57], 6, 7, No, coefficients_require_division_by_k_factorial)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0095","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d298b12839a29809870c262a7810f3223afd7174623fba018c7a90b5041df79","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122095,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":8,\"center\":7,\"derivatives\":[20,40,60],\"mode\":\"remainder_claim\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=remainder_claim, supplied derivatives=[20, 40, 60], center=7, order=2, alternate_center=8. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(remainder_claim, 2, [20, 40, 60], 7, 8, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [20, 40, 60], 7, 8, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0096","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e058cbc9f4eacec71389c7f13599659fbbda0e569c055f5868c5f14ea6a95c90","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122096,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":9,\"center\":8,\"derivatives\":[21,42],\"mode\":\"missing_derivative\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_derivative, supplied derivatives=[21, 42], center=8, order=2, alternate_center=9. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_derivative, 2, [21, 42], 8, 9, No, all_derivatives_zero_through_n_are_required)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [21, 42], 8, 9, No, all_derivatives_zero_through_n_are_required)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0097","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae8c24990f70e705e375cbbc19035f60c9ebdab41cc4fc2b9ee97fa47a561fe0","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122097,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":10,\"center\":9,\"derivatives\":[22,44,66],\"mode\":\"wrong_center\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=wrong_center, supplied derivatives=[22, 44, 66], center=9, order=2, alternate_center=10. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(wrong_center, 2, [22, 44, 66], 9, 10, No, a_jet_is_bound_to_its_declared_center)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [22, 44, 66], 9, 10, No, a_jet_is_bound_to_its_declared_center)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0098","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f54fedbdbe49d4eac232226d1983b0c53cf34b3c8728530153e50097cfd00da","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122098,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":11,\"center\":10,\"derivatives\":[23,46,69],\"mode\":\"missing_factorial\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_factorial, supplied derivatives=[23, 46, 69], center=10, order=2, alternate_center=11. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_factorial, 2, [23, 46, 69], 10, 11, No, coefficients_require_division_by_k_factorial)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_factorial, 2, [23, 46, 69], 10, 11, No, coefficients_require_division_by_k_factorial)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0099","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52eb12926024740f9f071fbd5ff649ea571009896bdd8ce801e429476d4e84b5","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122099,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":12,\"center\":11,\"derivatives\":[24,48,72],\"mode\":\"remainder_claim\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=remainder_claim, supplied derivatives=[24, 48, 72], center=11, order=2, alternate_center=12. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(remainder_claim, 2, [24, 48, 72], 11, 12, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (remainder_claim, 2, [24, 48, 72], 11, 12, No, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r02-taylor-polynomial-0100","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84ff6707306bebacc823278624f53fc24b4b67cc80debaff43558ee81118eae","family_id":"qf-mod43-taylor-definition-boundary","seed":202611122100,"semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":13,\"center\":12,\"derivatives\":[25,50],\"mode\":\"missing_derivative\",\"order\":2}"},"fields":{"title":"Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries","stem":"Use mode=missing_derivative, supplied derivatives=[25, 50], center=12, order=2, alternate_center=13. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).","answer":{"kind":"expression","value":"(missing_derivative, 2, [25, 50], 12, 13, No, all_derivatives_zero_through_n_are_required)","explanation":"zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (missing_derivative, 2, [25, 50], 12, 13, No, all_derivatives_zero_through_n_are_required)."},"skills":["Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}