{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0001","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1bad25830071e1e8e25a0ddaecbeaf3c382085f8b9d491aac2b6157c0ea8423","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196001,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":1,\"b\":1,\"p\":0,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=1*n^0, g(n)=1*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(1, 0, 1, 0, Yes, Yes, Yes, [1, 1], [1/1, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 0, 1, 0, Yes, Yes, Yes, [1, 1], [1/1, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0002","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5a28368ae945bafed72c3d2df9bcf24b929ff3056230ff3703c957d542e1c1a","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196002,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":2,\"b\":2,\"p\":1,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=2*n^1, g(n)=2*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(2, 1, 2, 3, Yes, No, No, [2, 1], does-not-exist)","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 1, 2, 3, Yes, No, No, [2, 1], does-not-exist)."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0003","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ee6a04c8b109c4ba8ac6f8072b487ca057f1cd67d4e5d19db97f2635ae7717d","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196003,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":3,\"b\":3,\"p\":2,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=3*n^2, g(n)=3*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(3, 2, 3, 0, No, Yes, No, does-not-exist, [1/3, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 2, 3, 0, No, Yes, No, does-not-exist, [1/3, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0004","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9766bafe56925fcf41d415668e7846d80d816bd0c5f833362d4599ad0d4fc635","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196004,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":4,\"b\":4,\"p\":3,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=4*n^3, g(n)=4*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(4, 3, 4, 3, Yes, Yes, Yes, [4, 1], [1/4, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, 4, 3, Yes, Yes, Yes, [4, 1], [1/4, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0005","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb4288cafa9f0cfc0b8fe9a2d25b36ff874544669bad2782dccbb62945f7f761","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196005,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":5,\"b\":5,\"p\":4,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=5*n^4, g(n)=5*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(5, 4, 5, 0, No, Yes, No, does-not-exist, [1/5, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 4, 5, 0, No, Yes, No, does-not-exist, [1/5, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0006","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbb2428405cf87edc999e443c02a0c60d145c74cf9072afa4bfceed0c8c0388b","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196006,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":6,\"b\":6,\"p\":5,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=6*n^5, g(n)=6*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(6, 5, 6, 3, No, Yes, No, does-not-exist, [1/6, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 5, 6, 3, No, Yes, No, does-not-exist, [1/6, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0007","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44c644782540fe4594d6b1d85071ce1340fde53416139023d58335ac824c78ed","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196007,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":7,\"b\":7,\"p\":0,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=7*n^0, g(n)=7*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(7, 0, 7, 0, Yes, Yes, Yes, [7, 1], [1/7, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 0, 7, 0, Yes, Yes, Yes, [7, 1], [1/7, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0008","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84349871e6e8d3b80c370646a6ae0a22241c7ee39283f67cddab3e4d752fb800","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196008,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":8,\"b\":1,\"p\":1,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=8*n^1, g(n)=1*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(8, 1, 1, 3, Yes, No, No, [8, 1], does-not-exist)","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 1, 1, 3, Yes, No, No, [8, 1], does-not-exist)."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0009","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8df436e54906daf9d626ddf8b5e53cd95d4ae80d07e64eb8fd465fa7dcd2db1","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196009,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":9,\"b\":2,\"p\":2,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=9*n^2, g(n)=2*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(9, 2, 2, 0, No, Yes, No, does-not-exist, [1/2, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 2, 2, 0, No, Yes, No, does-not-exist, [1/2, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0010","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cdcf74f27c681abb9c7a31afc5bea8a1e5a341f450c544760d74ccf81d68500","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196010,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":10,\"b\":3,\"p\":3,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=10*n^3, g(n)=3*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(10, 3, 3, 3, Yes, Yes, Yes, [10, 1], [1/3, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 3, 3, 3, Yes, Yes, Yes, [10, 1], [1/3, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0011","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1eb0ac5cc177ccf33fc21d1d600a948db36dc6132817026740c76425dd68489d","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196011,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":11,\"b\":4,\"p\":4,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=11*n^4, g(n)=4*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(11, 4, 4, 0, No, Yes, No, does-not-exist, [1/4, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, 4, 4, 0, No, Yes, No, does-not-exist, [1/4, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0012","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d8a17c8e5c7650ad7ce11ffccac615ae0b9f3dd33d34bd24e44f198fc3cea72","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196012,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":12,\"b\":5,\"p\":5,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=12*n^5, g(n)=5*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(12, 5, 5, 3, No, Yes, No, does-not-exist, [1/5, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, 5, 5, 3, No, Yes, No, does-not-exist, [1/5, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0013","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7d274e91303e4076622f7ed5083d787d5a3699bbd4bf54a4d144e42e55cc4f2","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196013,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":13,\"b\":6,\"p\":0,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=13*n^0, g(n)=6*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(13, 0, 6, 0, Yes, Yes, Yes, [13, 1], [1/6, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, 0, 6, 0, Yes, Yes, Yes, [13, 1], [1/6, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0014","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3219c0be7f52aabdeac9897d29e0da1b2a80f54d8669dd47893b646c80e870f2","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196014,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":14,\"b\":7,\"p\":1,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=14*n^1, g(n)=7*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(14, 1, 7, 3, Yes, No, No, [14, 1], does-not-exist)","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, 1, 7, 3, Yes, No, No, [14, 1], does-not-exist)."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0015","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dee5f42684f725d29e4b6b0b9593493a6f52235323c0b950ef65088de34f23d7","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196015,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":15,\"b\":1,\"p\":2,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=15*n^2, g(n)=1*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(15, 2, 1, 0, No, Yes, No, does-not-exist, [1/1, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 2, 1, 0, No, Yes, No, does-not-exist, [1/1, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0016","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2c606d5ab82f9f3fceaad9e31f2d8f8ee673dbb9af3fb76656f1f22a562ce73","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196016,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":16,\"b\":2,\"p\":3,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=16*n^3, g(n)=2*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(16, 3, 2, 3, Yes, Yes, Yes, [16, 1], [1/2, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, 3, 2, 3, Yes, Yes, Yes, [16, 1], [1/2, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0017","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26c4cdf56827c2afd4debfb424b8e93d5f02d9f97bd26526e5ea7abccc8275a7","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196017,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":17,\"b\":3,\"p\":4,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=17*n^4, g(n)=3*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(17, 4, 3, 0, No, Yes, No, does-not-exist, [1/3, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, 4, 3, 0, No, Yes, No, does-not-exist, [1/3, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0018","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eecb6891028b104e45388461840b64d0160d7fe90e0e8ec5b80b75006118ed4","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196018,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":18,\"b\":4,\"p\":5,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=18*n^5, g(n)=4*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(18, 5, 4, 3, No, Yes, No, does-not-exist, [1/4, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, 5, 4, 3, No, Yes, No, does-not-exist, [1/4, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0019","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39f5cbbfd1d16db17df811dc1d35ddf8ed13fe3dd8fcff11fc46666ff937ffb1","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196019,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":19,\"b\":5,\"p\":0,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=19*n^0, g(n)=5*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(19, 0, 5, 0, Yes, Yes, Yes, [19, 1], [1/5, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, 0, 5, 0, Yes, Yes, Yes, [19, 1], [1/5, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0020","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"407fab3a35b6e2cf88a2c07edb5576ea9c4705828c5952c0aab7b663997ab2c0","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196020,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":20,\"b\":6,\"p\":1,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=20*n^1, g(n)=6*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(20, 1, 6, 3, Yes, No, No, [20, 1], does-not-exist)","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 1, 6, 3, Yes, No, No, [20, 1], does-not-exist)."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0021","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0c45e007c4c0939516949b1648dc43a6ac96ab55bcda1dd902b13bd645e4b88","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196021,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":21,\"b\":7,\"p\":2,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=21*n^2, g(n)=7*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(21, 2, 7, 0, No, Yes, No, does-not-exist, [1/7, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, 2, 7, 0, No, Yes, No, does-not-exist, [1/7, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0022","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecd36ef6180a09c942ea8d3b16d391ea6621b3a300c83dbbb89732aa5dbc8824","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196022,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":22,\"b\":1,\"p\":3,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=22*n^3, g(n)=1*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(22, 3, 1, 3, Yes, Yes, Yes, [22, 1], [1/1, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 3, 1, 3, Yes, Yes, Yes, [22, 1], [1/1, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0023","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54e41ff8648bdf70951a466b015e097b2f403c946e2d4ecd480489f5266fdad1","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196023,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":23,\"b\":2,\"p\":4,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=23*n^4, g(n)=2*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(23, 4, 2, 0, No, Yes, No, does-not-exist, [1/2, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, 4, 2, 0, No, Yes, No, does-not-exist, [1/2, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0024","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aacc6ae05f52230a4c48996c13b49b911c3fba0d7dfe1aaf4a85017a63595b4a","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196024,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":24,\"b\":3,\"p\":5,\"q\":3}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=24*n^5, g(n)=3*n^3, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(24, 5, 3, 3, No, Yes, No, does-not-exist, [1/3, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 5, 3, 3, No, Yes, No, does-not-exist, [1/3, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0025","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24de26ce99c3365d5e8bcbe2b893dc2cdb13b7c5a1202d32e33d76f067121335","family_id":"qf-ext19-asymptotic-monomial-class","seed":202610196025,"semantic_template":"qf-ext19-asymptotic-monomial-class/{\"a\":25,\"b\":4,\"p\":0,\"q\":0}"},"fields":{"title":"Positive-Coefficient Monomial Tail Classes and Witnesses","stem":"Fix a cost model and explicit nonnegative functions f(n)=25*n^0, g(n)=4*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.","answer":{"kind":"expression","value":"(25, 0, 4, 0, Yes, Yes, Yes, [25, 1], [1/4, 1])","explanation":"Use the declared monomial family and compare exponents; retain the prescribed witnesses rather than substituting tighter constants or reducing 1/b. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (25, 0, 4, 0, Yes, Yes, Yes, [25, 1], [1/4, 1])."},"skills":["Positive-Coefficient Monomial Tail Classes and Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0026","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c60bd4297bacf6aa28ec404d10b2d4c401c998997e32e2d7b78b737cfea57f66","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196026,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[0,1,0],\"coeffs_g\":[1,1,2,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[0, 1, 0], g=[1, 1, 2, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([0, 1, 0], [1, 1, 2, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0], [1, 1, 2, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0027","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3083e6ce781538575e073bc0ac07844292705e22fae0791774fab3c7264bbb20","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196027,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[1,2,1],\"coeffs_g\":[1,2,0,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[1, 2, 1], g=[1, 2, 0, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([1, 2, 1], [1, 2, 0, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1], [1, 2, 0, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0028","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86de1c53e1bb102de944b09195da0fdaf58cafc4119a96a38f1bd2dc961e9da8","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196028,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[2,3,0],\"coeffs_g\":[1,3,1,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[2, 3, 0], g=[1, 3, 1, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([2, 3, 0], [1, 3, 1, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 0], [1, 3, 1, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0029","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d1476854b3a3fca7667409f29d3ac4ea59af546161752b3391baafd14f46cfc","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196029,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[3,1,1],\"coeffs_g\":[1,4,2,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[3, 1, 1], g=[1, 4, 2, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([3, 1, 1], [1, 4, 2, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 1], [1, 4, 2, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0030","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dd46d4e9b3e77ac06430e816819661602e54412d0077a29f2e5bf479df383b0","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196030,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[4,2,0],\"coeffs_g\":[1,5,0,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[4, 2, 0], g=[1, 5, 0, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([4, 2, 0], [1, 5, 0, 0], 1, 1, Yes, Yes, Yes)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2, 0], [1, 5, 0, 0], 1, 1, Yes, Yes, Yes)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0031","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61cec06ed97960edf88a4e86b54faee7500766041003f070c56aff59ae309d67","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196031,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[5,3,1],\"coeffs_g\":[1,6,1,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[5, 3, 1], g=[1, 6, 1, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([5, 3, 1], [1, 6, 1, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3, 1], [1, 6, 1, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0032","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02bc6960df30326e0fe7dfa247c3d7e1bc60d3a26538b2e58b198f314eba6cf6","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196032,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[6,1,0],\"coeffs_g\":[1,7,2,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[6, 1, 0], g=[1, 7, 2, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([6, 1, 0], [1, 7, 2, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 1, 0], [1, 7, 2, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0033","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80e1454f18dea61794496714c0d3db86800877e3cf82f3530f0640cc224832da","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196033,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[7,2,1],\"coeffs_g\":[1,8,0,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[7, 2, 1], g=[1, 8, 0, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([7, 2, 1], [1, 8, 0, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 2, 1], [1, 8, 0, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0034","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d87c1a0f0ee6a077308c234db75272b0b59ba239056cd54bb7a5ba6cb6887194","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196034,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[8,3,0],\"coeffs_g\":[1,9,1,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[8, 3, 0], g=[1, 9, 1, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([8, 3, 0], [1, 9, 1, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 3, 0], [1, 9, 1, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0035","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"076388a4e0d4fb4d3980a5111d92a5079733ac8e66317cd477f8dcf29b179fa3","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196035,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[9,1,1],\"coeffs_g\":[1,10,2,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[9, 1, 1], g=[1, 10, 2, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([9, 1, 1], [1, 10, 2, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 1, 1], [1, 10, 2, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0036","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"417cd997d09d60145b6e5e160a0af065dda62080c815200f190374bc467b3103","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196036,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[10,2,0],\"coeffs_g\":[1,11,0,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[10, 2, 0], g=[1, 11, 0, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([10, 2, 0], [1, 11, 0, 0], 1, 1, Yes, Yes, Yes)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2, 0], [1, 11, 0, 0], 1, 1, Yes, Yes, Yes)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0037","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"345e727fa823eedc1a8cf94c18169ddd56f9bf2df1a86b418e473879fc437577","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196037,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[11,3,1],\"coeffs_g\":[1,12,1,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[11, 3, 1], g=[1, 12, 1, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([11, 3, 1], [1, 12, 1, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 3, 1], [1, 12, 1, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0038","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9d2ec8d686f69c4ff21c1577ae7db81cd6edb4bbf62eaa9b5e046eaeb790155","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196038,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[12,1,0],\"coeffs_g\":[1,13,2,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[12, 1, 0], g=[1, 13, 2, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([12, 1, 0], [1, 13, 2, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 1, 0], [1, 13, 2, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0039","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"509f955566ac97bed191c2d28cd1c54cbacfdd8f0172d9de38a2512e6e51debe","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196039,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[13,2,1],\"coeffs_g\":[1,14,0,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[13, 2, 1], g=[1, 14, 0, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([13, 2, 1], [1, 14, 0, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([13, 2, 1], [1, 14, 0, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0040","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55a19507b24573b93efc8b1750505d309e046dc523bda25f671eb605e2d8e8cd","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196040,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[14,3,0],\"coeffs_g\":[1,15,1,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[14, 3, 0], g=[1, 15, 1, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([14, 3, 0], [1, 15, 1, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([14, 3, 0], [1, 15, 1, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0041","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b97f08484e1db9befc053aa35fde45165dcdb0e820587703768766b3b0d277ec","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196041,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[15,1,1],\"coeffs_g\":[1,16,2,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[15, 1, 1], g=[1, 16, 2, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([15, 1, 1], [1, 16, 2, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([15, 1, 1], [1, 16, 2, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0042","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7163030daa0980af26fb82ccb605ebfd6ff6b1a82a00fb8c747eb43717850a10","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196042,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[16,2,0],\"coeffs_g\":[1,17,0,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[16, 2, 0], g=[1, 17, 0, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([16, 2, 0], [1, 17, 0, 0], 1, 1, Yes, Yes, Yes)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([16, 2, 0], [1, 17, 0, 0], 1, 1, Yes, Yes, Yes)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0043","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a4aa8a342a61a800a285e1458ffc79161612b8f4a7964b21a4bfa704997331b","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196043,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[17,3,1],\"coeffs_g\":[1,18,1,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[17, 3, 1], g=[1, 18, 1, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([17, 3, 1], [1, 18, 1, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([17, 3, 1], [1, 18, 1, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0044","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"745c2d4899b3908095d3caeea8c586694770249298974b66da1806d16213d180","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196044,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[18,1,0],\"coeffs_g\":[1,19,2,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[18, 1, 0], g=[1, 19, 2, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([18, 1, 0], [1, 19, 2, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([18, 1, 0], [1, 19, 2, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0045","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56bad32d8ec61a8958e45f26caff6cb627fbfd1bbde6205e83d98b0978212e6a","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196045,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[19,2,1],\"coeffs_g\":[1,20,0,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[19, 2, 1], g=[1, 20, 0, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([19, 2, 1], [1, 20, 0, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([19, 2, 1], [1, 20, 0, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0046","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff5b1dcf156ce15c45a30045ca62849319971fc8bf1e2db089b44012ee56908","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196046,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[20,3,0],\"coeffs_g\":[1,21,1,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[20, 3, 0], g=[1, 21, 1, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([20, 3, 0], [1, 21, 1, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([20, 3, 0], [1, 21, 1, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0047","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf9f4e1fbdc72687f2dad7e9976e62beda75adf7e0b8a8cbdf15fb526eb9d330","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196047,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[21,1,1],\"coeffs_g\":[1,22,2,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[21, 1, 1], g=[1, 22, 2, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([21, 1, 1], [1, 22, 2, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([21, 1, 1], [1, 22, 2, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0048","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24dd2d831e218cf00f9e3fa6ed288eb9791ec312c254f6e9fe66c6cd1aa3b795","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196048,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[22,2,0],\"coeffs_g\":[1,23,0,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[22, 2, 0], g=[1, 23, 0, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([22, 2, 0], [1, 23, 0, 0], 1, 1, Yes, Yes, Yes)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([22, 2, 0], [1, 23, 0, 0], 1, 1, Yes, Yes, Yes)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0049","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de8534964b0aeb6749f2bb2ffb0ecdff7d2b17a255ea0d7790c18b15ec8af56d","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196049,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[23,3,1],\"coeffs_g\":[1,24,1,1]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[23, 3, 1], g=[1, 24, 1, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([23, 3, 1], [1, 24, 1, 1], 2, 3, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([23, 3, 1], [1, 24, 1, 1], 2, 3, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0050","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1798e3cfbb055ca0cc0ef9bc30ce783e262b445dd36b3e1bcce35a416ec0650e","family_id":"qf-ext19-asymptotic-polynomial-degree","seed":202610196050,"semantic_template":"qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[24,1,0],\"coeffs_g\":[1,25,2,0]}"},"fields":{"title":"Nonnegative Polynomial Degrees Including Both Zero Polynomials","stem":"Fix the cost model. Coefficients in ascending powers are f=[24, 1, 0], g=[1, 25, 2, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.","answer":{"kind":"expression","value":"([24, 1, 0], [1, 25, 2, 0], 1, 2, Yes, No, No)","explanation":"Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([24, 1, 0], [1, 25, 2, 0], 1, 2, Yes, No, No)."},"skills":["Nonnegative Polynomial Degrees Including Both Zero Polynomials"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0051","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8e5b87d92c61b1b21cec8cdf42200d0229a46b951072f776dbe924ef6bda5db","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196051,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":0,\"b\":1}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=0, b=1, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(0, 1, 1, 0, Yes, 0n+1≤1(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 1, 0, Yes, 0n+1≤1(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0052","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"733f754bbb64f747bc91c27efe3e4622b497b28ddf264a8bffa5642514db334b","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196052,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":1,\"b\":3}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=1, b=3, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(1, 3, 4, 0, Yes, 1n+3≤4(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 4, 0, Yes, 1n+3≤4(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0053","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"386ac043da1fc12bede00adad77a71cb3bf310c0a6fd1f33b545425ad10f89d6","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196053,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":2,\"b\":5}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=2, b=5, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(2, 5, 7, 0, Yes, 2n+5≤7(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 7, 0, Yes, 2n+5≤7(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0054","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dd86b36c8a265be883dc6aa3d808b9251f4d00c0675f2e99434a8ff6e1ebea6","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196054,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":3,\"b\":7}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=3, b=7, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(3, 7, 10, 0, Yes, 3n+7≤10(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 7, 10, 0, Yes, 3n+7≤10(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0055","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d9e864fb45bbc159e96bcf79f54716df2e790a9d3a842a2dba1a8baac9de0d","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196055,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":4,\"b\":9}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=4, b=9, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(4, 9, 13, 0, Yes, 4n+9≤13(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 9, 13, 0, Yes, 4n+9≤13(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0056","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3de4022c839aa8be055faa7809fd6375f19bdb296135e58efe874f9c4934a5d","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196056,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":5,\"b\":11}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=5, b=11, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(5, 11, 16, 0, Yes, 5n+11≤16(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 11, 16, 0, Yes, 5n+11≤16(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0057","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c88c4e54c7c156993e951f087939f5d00dcef1a6ecc5a2c26673b845ebb88d2c","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196057,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":6,\"b\":13}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=6, b=13, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(6, 13, 19, 0, Yes, 6n+13≤19(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 13, 19, 0, Yes, 6n+13≤19(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0058","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4401185390fb42d7426834b00bfbc624da7536ef85d0aecdea94a5a09592a9e2","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196058,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":7,\"b\":15}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=7, b=15, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(7, 15, 22, 0, Yes, 7n+15≤22(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 15, 22, 0, Yes, 7n+15≤22(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0059","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ccc9471b86b8cdb3eeddf077f71b4324cbf187abe2f7fde0b714f880e8825a6","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196059,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":8,\"b\":17}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=8, b=17, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(8, 17, 25, 0, Yes, 8n+17≤25(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 17, 25, 0, Yes, 8n+17≤25(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0060","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16d53616da6d8258ae981228e126a093f999f207d0c891007032ad35fdefe45b","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196060,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":9,\"b\":19}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=9, b=19, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(9, 19, 28, 0, Yes, 9n+19≤28(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 19, 28, 0, Yes, 9n+19≤28(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0061","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e55ed9cf866d1beeca997f75ee01e64b16ce254574e94aff900ebea16fe7b68f","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196061,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":10,\"b\":21}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=10, b=21, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(10, 21, 31, 0, Yes, 10n+21≤31(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 21, 31, 0, Yes, 10n+21≤31(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0062","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"889de91a1858b548daf8275f99182a09c39d19510f2922365af9084c382da7ef","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196062,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":11,\"b\":23}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=11, b=23, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(11, 23, 34, 0, Yes, 11n+23≤34(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, 23, 34, 0, Yes, 11n+23≤34(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0063","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b8ca0ed5580b9a94f12f73736678d42569fa29f00a179132d7f17d301ad00d4","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196063,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":12,\"b\":25}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=12, b=25, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(12, 25, 37, 0, Yes, 12n+25≤37(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, 25, 37, 0, Yes, 12n+25≤37(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0064","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"354d0f6c49628996fee857ab837dd5fd3b7e5de38d3b938dc7424e688db65463","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196064,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":13,\"b\":27}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=13, b=27, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(13, 27, 40, 0, Yes, 13n+27≤40(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, 27, 40, 0, Yes, 13n+27≤40(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0065","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebe90f74501d46fe5235cc7e99354832bd2727572f95ff806ad3e96806875ae7","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196065,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":14,\"b\":29}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=14, b=29, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(14, 29, 43, 0, Yes, 14n+29≤43(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, 29, 43, 0, Yes, 14n+29≤43(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0066","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe009f8919fc899119be92d49b04e9bfd20fbfd832c70e9bf19d7f383760ee03","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196066,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":15,\"b\":31}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=15, b=31, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(15, 31, 46, 0, Yes, 15n+31≤46(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 31, 46, 0, Yes, 15n+31≤46(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0067","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3f29e0880208d2d75c30c08667649beaf8e67af9c2b0b4f56845106c3f757f8","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196067,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":16,\"b\":33}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=16, b=33, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(16, 33, 49, 0, Yes, 16n+33≤49(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, 33, 49, 0, Yes, 16n+33≤49(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0068","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dbcb8f46eb3529def808222a01c06314391417e2db5bd567303165457f8ccdc","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196068,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":17,\"b\":35}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=17, b=35, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(17, 35, 52, 0, Yes, 17n+35≤52(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, 35, 52, 0, Yes, 17n+35≤52(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0069","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8f9f90285064fc921edf167d0a0da4138a3f6add84aef38c8ad9b1e29e75610","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196069,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":18,\"b\":37}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=18, b=37, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(18, 37, 55, 0, Yes, 18n+37≤55(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, 37, 55, 0, Yes, 18n+37≤55(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0070","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"513a27388311ac6e4f554460e38c9cc2b5c6c4bc6c6209533595cd16be82f00e","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196070,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":19,\"b\":39}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=19, b=39, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(19, 39, 58, 0, Yes, 19n+39≤58(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, 39, 58, 0, Yes, 19n+39≤58(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0071","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81c871b8873ce9caa4fe6c0fccdb009ec87ab0853355e94e73a055ec27ffee34","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196071,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":20,\"b\":41}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=20, b=41, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(20, 41, 61, 0, Yes, 20n+41≤61(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 41, 61, 0, Yes, 20n+41≤61(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0072","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cce04420c1808a547f815065e9ea9f8fc5e58c9e99b6f7a6ac238e6d245bda43","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196072,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":21,\"b\":43}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=21, b=43, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(21, 43, 64, 0, Yes, 21n+43≤64(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, 43, 64, 0, Yes, 21n+43≤64(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0073","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5200c17faa46fe7400c43a8e49896e4a0c44f5a35170617d7a0e58ff81dd348","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196073,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":22,\"b\":45}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=22, b=45, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(22, 45, 67, 0, Yes, 22n+45≤67(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 45, 67, 0, Yes, 22n+45≤67(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0074","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a518640c2a041d36e10e3812bc9a57d38a028612363e1c44a0c9de042833d1e","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196074,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":23,\"b\":47}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=23, b=47, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(23, 47, 70, 0, Yes, 23n+47≤70(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, 47, 70, 0, Yes, 23n+47≤70(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0075","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d631c66dd1336585fe1e9af384ae7807e4808412fd388040f7463ac72888c8f3","family_id":"qf-ext19-asymptotic-linear-witness","seed":202610196075,"semantic_template":"qf-ext19-asymptotic-linear-witness/{\"a\":24,\"b\":49}"},"fields":{"title":"An Explicit All-Input Linear Resource Witness","stem":"Use natural n≥0, nonnegative integers a=24, b=49, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).","answer":{"kind":"expression","value":"(24, 49, 73, 0, Yes, 24n+49≤73(n+1) for all n≥0)","explanation":"The specified nonnegative-coefficient bound is an all-input algebraic witness, not a conclusion inferred from a few measured points. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 49, 73, 0, Yes, 24n+49≤73(n+1) for all n≥0)."},"skills":["An Explicit All-Input Linear Resource Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0076","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2276c2e2acf95f30513dde4d64e2683fadc685bfaa056d18fcf912a3dd3aa301","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196076,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":0,\"p\":0,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^0, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,0]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(0, 1, 0, [[0, 1, 0]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 0, [[0, 1, 0]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0077","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87ecdb24db234f9e1e206e6311fc275dbf340816ab1eac7fc9a9a4dba02ffc86","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196077,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":1,\"p\":1,\"q\":3}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^1, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,1]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(1, 3, 1, [[0, 0, 0], [1, 1, 1]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 1, [[0, 0, 0], [1, 1, 1]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0078","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a07cf50b1e80c2bc5d4b738b328a88d92abf750c5f720b0844cb607383edd84d","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196078,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":2,\"p\":2,\"q\":5}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^2, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,2]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(2, 5, 2, [[0, 0, 0], [1, 1, 1], [2, 4, 32]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 2, [[0, 0, 0], [1, 1, 1], [2, 4, 32]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0079","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dc0a0116d65f7475dc7bfbd249ee2ddd4a1e9916b0ad61f22c3058a864a8e0e","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196079,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":3,\"p\":3,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^3, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,3]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(3, 1, 3, [[0, 0, 0], [1, 1, 1], [2, 8, 2], [3, 27, 3]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 1, 3, [[0, 0, 0], [1, 1, 1], [2, 8, 2], [3, 27, 3]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0080","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef4c418ec8029360542095ff722dc0761aa46cfe95c8ac87d6847a98312d79a1","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196080,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":4,\"p\":4,\"q\":3}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^4, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,4]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(4, 3, 4, [[0, 0, 0], [1, 1, 1], [2, 16, 8], [3, 81, 27], [4, 256, 64]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, 4, [[0, 0, 0], [1, 1, 1], [2, 16, 8], [3, 81, 27], [4, 256, 64]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0081","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f854950938f2879e946fa3aa9c4263e9994f5a923a9c18598218f56d1abefb67","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196081,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":5,\"p\":5,\"q\":5}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,5]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(5, 5, 5, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024], [5, 3125, 3125]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 5, 5, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024], [5, 3125, 3125]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0082","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa6ff723834dd9844bb58c83502b9ac5c425512bc1e54ca2a869f1b400ef7861","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196082,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":6,\"p\":0,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^0, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,6]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(0, 1, 6, [[0, 1, 0], [1, 1, 1], [2, 1, 2], [3, 1, 3], [4, 1, 4], [5, 1, 5], [6, 1, 6]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 6, [[0, 1, 0], [1, 1, 1], [2, 1, 2], [3, 1, 3], [4, 1, 4], [5, 1, 5], [6, 1, 6]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0083","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e67ab9e05e41174e7bf418e7b365012af1d3e901aa9ec77b890b6511d927531b","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196083,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":0,\"p\":1,\"q\":3}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^1, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,0]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(1, 3, 0, [[0, 0, 0]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 0, [[0, 0, 0]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0084","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"052a5a1608e2fac144c1f68dc41f314729819e952eaa4ccda76f83752f2d3e58","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196084,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":1,\"p\":2,\"q\":5}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^2, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,1]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(2, 5, 1, [[0, 0, 0], [1, 1, 1]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 1, [[0, 0, 0], [1, 1, 1]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0085","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb2235ec9acf53c162564bc3c1774d2fbb03274ce03844ba111182b32130b357","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196085,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":2,\"p\":3,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^3, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,2]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(3, 1, 2, [[0, 0, 0], [1, 1, 1], [2, 8, 2]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 1, 2, [[0, 0, 0], [1, 1, 1], [2, 8, 2]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0086","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21824957876ae3b1316c7490dd10f61e588979b08828a2ac5cf1b887942eaa94","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196086,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":3,\"p\":4,\"q\":3}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^4, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,3]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(4, 3, 3, [[0, 0, 0], [1, 1, 1], [2, 16, 8], [3, 81, 27]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, 3, [[0, 0, 0], [1, 1, 1], [2, 16, 8], [3, 81, 27]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0087","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63d27b1aa4fb3fba395349ab21756a1d334662fcf1dc3a8b72f2b90b0c2a813c","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196087,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":4,\"p\":5,\"q\":5}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,4]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(5, 5, 4, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 5, 4, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0088","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1df627357d8b8159c9e16150f375936fa4c6e2d6520739df53b75c078542f327","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196088,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":5,\"p\":0,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^0, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,5]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(0, 1, 5, [[0, 1, 0], [1, 1, 1], [2, 1, 2], [3, 1, 3], [4, 1, 4], [5, 1, 5]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 5, [[0, 1, 0], [1, 1, 1], [2, 1, 2], [3, 1, 3], [4, 1, 4], [5, 1, 5]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0089","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6972741cbbc5b24b54505ee9f66e7f925f8825fb80d2a5c0bc9419d8085df42","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196089,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":6,\"p\":1,\"q\":3}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^1, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,6]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(1, 3, 6, [[0, 0, 0], [1, 1, 1], [2, 2, 8], [3, 3, 27], [4, 4, 64], [5, 5, 125], [6, 6, 216]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 6, [[0, 0, 0], [1, 1, 1], [2, 2, 8], [3, 3, 27], [4, 4, 64], [5, 5, 125], [6, 6, 216]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0090","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a063fe3a0be21785f55bd8ecb1d55766018c1b12207ceaf3a8e7659c1abc345","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196090,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":0,\"p\":2,\"q\":5}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^2, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,0]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(2, 5, 0, [[0, 0, 0]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 0, [[0, 0, 0]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0091","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5752abcffa6cf0a16895b76c9d5098b3df1b3f707860f3efb4347a8f9dfd3a0","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196091,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":1,\"p\":3,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^3, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,1]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(3, 1, 1, [[0, 0, 0], [1, 1, 1]], Yes, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 1, 1, [[0, 0, 0], [1, 1, 1]], Yes, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0092","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec27a430121c76d565e10b6ef8d28833171a317d79c07394e3b30ff9d96f9209","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196092,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":2,\"p\":4,\"q\":3}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^4, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,2]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(4, 3, 2, [[0, 0, 0], [1, 1, 1], [2, 16, 8]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, 2, [[0, 0, 0], [1, 1, 1], [2, 16, 8]], No, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0093","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"590058d57afbb23d8c04d262164cf9ecfcbde8eb353d3b12e4a6d14cf1e338b0","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196093,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":3,\"p\":5,\"q\":5}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,3]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(5, 5, 3, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 5, 3, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0094","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcfd150262355484af75756286e5034aa417931dbb75ac2f17e8c81b933adbf0","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196094,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":4,\"p\":0,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^0, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,4]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(0, 1, 4, [[0, 1, 0], [1, 1, 1], [2, 1, 2], [3, 1, 3], [4, 1, 4]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 4, [[0, 1, 0], [1, 1, 1], [2, 1, 2], [3, 1, 3], [4, 1, 4]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0095","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed52b287197f7392beacdfd36716a346b463515b3d84c0a281e73d1331db18ae","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196095,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":5,\"p\":1,\"q\":3}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^1, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,5]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(1, 3, 5, [[0, 0, 0], [1, 1, 1], [2, 2, 8], [3, 3, 27], [4, 4, 64], [5, 5, 125]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 5, [[0, 0, 0], [1, 1, 1], [2, 2, 8], [3, 3, 27], [4, 4, 64], [5, 5, 125]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0096","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3af67e13f62f8d34018883b9bcbd308764e55a8ab47ac33109dcee3308028032","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196096,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":6,\"p\":2,\"q\":5}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^2, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,6]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(2, 5, 6, [[0, 0, 0], [1, 1, 1], [2, 4, 32], [3, 9, 243], [4, 16, 1024], [5, 25, 3125], [6, 36, 7776]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 6, [[0, 0, 0], [1, 1, 1], [2, 4, 32], [3, 9, 243], [4, 16, 1024], [5, 25, 3125], [6, 36, 7776]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0097","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3034f486335ef8978a83f8fcde6fe15063692b699a4df24e4bab82114cd81561","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196097,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":0,\"p\":3,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^3, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,0]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(3, 1, 0, [[0, 0, 0]], Yes, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 1, 0, [[0, 0, 0]], Yes, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0098","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b5046696d2258da623385c004d97031777611a447ae94d1a3d2506d5232dfa","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196098,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":1,\"p\":4,\"q\":3}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^4, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,1]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(4, 3, 1, [[0, 0, 0], [1, 1, 1]], Yes, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, 1, [[0, 0, 0], [1, 1, 1]], Yes, [No, Yes, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0099","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbc69b35da3f3a7acb60d6578846ce74c4d5f83ceab5cd20b8446e758732ba3f","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196099,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":2,\"p\":5,\"q\":5}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,2]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(5, 5, 2, [[0, 0, 0], [1, 1, 1], [2, 32, 32]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 5, 2, [[0, 0, 0], [1, 1, 1], [2, 32, 32]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r06-asymptotic-resource-bound-0100","batch_id":"batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdc56f9404eefecf9e1672169d504c764fe422be5c450de49cec43179a59575f","family_id":"qf-ext19-asymptotic-benchmark-insufficiency","seed":202610196100,"semantic_template":"qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":3,\"p\":0,\"q\":1}"},"fields":{"title":"Finite Sample Results versus Independently Known Tail Classes","stem":"Fix f(n)=n^0, g(n)=n^1, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,3]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).","answer":{"kind":"expression","value":"(0, 1, 3, [[0, 1, 0], [1, 1, 1], [2, 1, 2], [3, 1, 3]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)","explanation":"Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 3, [[0, 1, 0], [1, 1, 1], [2, 1, 2], [3, 1, 3]], No, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof)."},"skills":["Finite Sample Results versus Independently Known Tail Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}