{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0001","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af6b11a3a0d4f4ba38fcb4ce340a19fcc0b2b8da7f107098c61c41c0648b04e0","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132001,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":5,\"x\":[1,2]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=1/2, cutoff=5, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(1/2, 5, [1, 1/2, 1/8, 1/48, 1/384, 1/3840], [1, 3/2, 13/8, 79/48, 211/128, 6331/3840], 6331/3840, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (1/2, 5, [1, 1/2, 1/8, 1/48, 1/384, 1/3840], [1, 3/2, 13/8, 79/48, 211/128, 6331/3840], 6331/3840, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0002","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38123da9d4427ec0cef4a6a093af076547fa4511f7316958c9c438dafb08b24c","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132002,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":6,\"x\":[2,3]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=2/3, cutoff=6, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(2/3, 6, [1, 2/3, 2/9, 4/81, 2/243, 4/3645, 4/32805], [1, 5/3, 17/9, 157/81, 473/243, 7099/3645, 12779/6561], 12779/6561, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (2/3, 6, [1, 2/3, 2/9, 4/81, 2/243, 4/3645, 4/32805], [1, 5/3, 17/9, 157/81, 473/243, 7099/3645, 12779/6561], 12779/6561, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0003","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cc4aff83f18d1127b66048d99255bcfbf269ad1cb6738734beaaa7d90266715","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132003,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":7,\"x\":[3,4]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=3/4, cutoff=7, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(3/4, 7, [1, 3/4, 9/32, 9/128, 27/2048, 81/40960, 81/327680, 243/9175040], [1, 7/4, 65/32, 269/128, 4331/2048, 86701/40960, 693689/327680, 3884707/1835008], 3884707/1835008, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (3/4, 7, [1, 3/4, 9/32, 9/128, 27/2048, 81/40960, 81/327680, 243/9175040], [1, 7/4, 65/32, 269/128, 4331/2048, 86701/40960, 693689/327680, 3884707/1835008], 3884707/1835008, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0004","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e39361ac82d98c00062396b95968c09e2ba024288536765a2c9317a752c2aef","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132004,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":8,\"x\":[4,5]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=4/5, cutoff=8, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(4/5, 8, [1, 4/5, 8/25, 32/375, 32/1875, 128/46875, 256/703125, 1024/24609375, 512/123046875], [1, 9/5, 53/25, 827/375, 1389/625, 104303/46875, 1564801/703125, 6085451/2734375, 273845807/123046875], 273845807/123046875, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (4/5, 8, [1, 4/5, 8/25, 32/375, 32/1875, 128/46875, 256/703125, 1024/24609375, 512/123046875], [1, 9/5, 53/25, 827/375, 1389/625, 104303/46875, 1564801/703125, 6085451/2734375, 273845807/123046875], 273845807/123046875, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0005","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08aaae32003954ee91b75ca09c9b6f0176e2737a22efc8db33628e2aed240b98","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132005,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":9,\"x\":[5,6]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=5/6, cutoff=9, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0006","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e910762bab19a045f2ebbded68d2a2430ce5e4d64a6d509e7b57ab5572b451a","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132006,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":10,\"x\":[6,7]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=6/7, cutoff=10, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(6/7, 10, [1, 6/7, 18/49, 36/343, 54/2401, 324/84035, 324/588245, 1944/28824005, 1458/201768035, 972/1412376245, 2916/49433168575], [1, 13/7, 109/49, 799/343, 5647/2401, 197969/84035, 1386107/588245, 67921187/28824005, 475449767/201768035, 3328149341/1412376245, 116485229851/49433168575], 116485229851/49433168575, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (6/7, 10, [1, 6/7, 18/49, 36/343, 54/2401, 324/84035, 324/588245, 1944/28824005, 1458/201768035, 972/1412376245, 2916/49433168575], [1, 13/7, 109/49, 799/343, 5647/2401, 197969/84035, 1386107/588245, 67921187/28824005, 475449767/201768035, 3328149341/1412376245, 116485229851/49433168575], 116485229851/49433168575, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0007","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"695b945eb411f5c864c077247cd646aaa19fba81a41e8c4239410c6754160bb7","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132007,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":11,\"x\":[7,8]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=7/8, cutoff=11, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(7/8, 11, [1, 7/8, 49/128, 343/3072, 2401/98304, 16807/3932160, 117649/188743680, 117649/1509949440, 823543/96636764160, 5764801/6957847019520, 40353607/556627761561600, 282475249/48983243017420800], [1, 15/8, 289/128, 7279/3072, 78443/32768, 9429967/3932160, 90551213/37748736, 1207388723/503316480, 231819458359/96636764160, 16691006766649/6957847019520, 445093527228509/185542587187200, 4700187658832065/1959329720696832], 4700187658832065/1959329720696832, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (7/8, 11, [1, 7/8, 49/128, 343/3072, 2401/98304, 16807/3932160, 117649/188743680, 117649/1509949440, 823543/96636764160, 5764801/6957847019520, 40353607/556627761561600, 282475249/48983243017420800], [1, 15/8, 289/128, 7279/3072, 78443/32768, 9429967/3932160, 90551213/37748736, 1207388723/503316480, 231819458359/96636764160, 16691006766649/6957847019520, 445093527228509/185542587187200, 4700187658832065/1959329720696832], 4700187658832065/1959329720696832, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0008","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"925c5bb609c57532c8c6498933b31eb7b728615fb7a2f1e55f7185c70eba5680","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132008,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":12,\"x\":[8,9]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=8/9, cutoff=12, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(8/9, 12, [1, 8/9, 32/81, 256/2187, 512/19683, 4096/885735, 16384/23914845, 131072/1506635235, 131072/13559717115, 1048576/1098337086315, 4194304/49425168884175, 33554432/4893091719533325, 67108864/132113476427399775], [1, 17/9, 185/81, 5251/2187, 47771/19683, 2153791/885735, 58168741/23914845, 732952351/301327047, 32982986867/13559717115, 2671622984803/1098337086315, 120223038510439/49425168884175, 11902080846087893/4893091719533325, 12854247316459279/5284539057095991], 12854247316459279/5284539057095991, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (8/9, 12, [1, 8/9, 32/81, 256/2187, 512/19683, 4096/885735, 16384/23914845, 131072/1506635235, 131072/13559717115, 1048576/1098337086315, 4194304/49425168884175, 33554432/4893091719533325, 67108864/132113476427399775], [1, 17/9, 185/81, 5251/2187, 47771/19683, 2153791/885735, 58168741/23914845, 732952351/301327047, 32982986867/13559717115, 2671622984803/1098337086315, 120223038510439/49425168884175, 11902080846087893/4893091719533325, 12854247316459279/5284539057095991], 12854247316459279/5284539057095991, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0009","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c2ae9251b7d31490037e008c9b7184097a8da4740a6056e159196d49a560c0f","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132009,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":13,\"x\":[9,10]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=9/10, cutoff=13, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(9/10, 13, [1, 9/10, 81/200, 243/2000, 2187/80000, 19683/4000000, 59049/80000000, 531441/5600000000, 4782969/448000000000, 4782969/4480000000000, 43046721/448000000000000, 387420489/49280000000000000, 1162261467/1971200000000000000, 10460353203/256256000000000000000], [1, 19/10, 461/200, 4853/2000, 196307/80000, 9835033/4000000, 196759709/80000000, 13773711071/5600000000, 1101901668649/448000000000, 11019021469459/4480000000000, 1101902189992621/448000000000000, 121209241286608799/49280000000000000, 4848369652626613427/1971200000000000000, 630288054851920098713/256256000000000000000], 630288054851920098713/256256000000000000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (9/10, 13, [1, 9/10, 81/200, 243/2000, 2187/80000, 19683/4000000, 59049/80000000, 531441/5600000000, 4782969/448000000000, 4782969/4480000000000, 43046721/448000000000000, 387420489/49280000000000000, 1162261467/1971200000000000000, 10460353203/256256000000000000000], [1, 19/10, 461/200, 4853/2000, 196307/80000, 9835033/4000000, 196759709/80000000, 13773711071/5600000000, 1101901668649/448000000000, 11019021469459/4480000000000, 1101902189992621/448000000000000, 121209241286608799/49280000000000000, 4848369652626613427/1971200000000000000, 630288054851920098713/256256000000000000000], 630288054851920098713/256256000000000000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0010","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32e257f235adb4ecb278e5fb7d4fbd97187b37985981fb71c5d359cae46bdcf9","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132010,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":14,\"x\":[10,11]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=10/11, cutoff=14, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(10/11, 14, [1, 10/11, 50/121, 500/3993, 1250/43923, 2500/483153, 12500/15944049, 125000/1227691773, 156250/13504609503, 1562500/1336956340797, 1562500/14706519748767, 15625000/1779488889600807, 39062500/58723133356826631, 390625000/8397408070026208233, 1953125000/646600421392018033941], [1, 21/11, 281/121, 9773/3993, 36251/14641, 1198783/483153, 39572339/15944049, 1015731701/409230591, 4788471769/1929229929, 3318412498417/1336956340797, 12167513015029/4902173249589, 4416807240080527/1779488889600807, 145754638961719891/58723133356826631, 6947637790638856471/2799136023342069411, 1604904329639528969801/646600421392018033941], 1604904329639528969801/646600421392018033941, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (10/11, 14, [1, 10/11, 50/121, 500/3993, 1250/43923, 2500/483153, 12500/15944049, 125000/1227691773, 156250/13504609503, 1562500/1336956340797, 1562500/14706519748767, 15625000/1779488889600807, 39062500/58723133356826631, 390625000/8397408070026208233, 1953125000/646600421392018033941], [1, 21/11, 281/121, 9773/3993, 36251/14641, 1198783/483153, 39572339/15944049, 1015731701/409230591, 4788471769/1929229929, 3318412498417/1336956340797, 12167513015029/4902173249589, 4416807240080527/1779488889600807, 145754638961719891/58723133356826631, 6947637790638856471/2799136023342069411, 1604904329639528969801/646600421392018033941], 1604904329639528969801/646600421392018033941, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0011","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3029cec683b32cdb0b6f06c767eafe4e3e4e627172d1f9050156d0a90f211a32","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132011,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":15,\"x\":[11,12]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=11/12, cutoff=15, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(11/12, 15, [1, 11/12, 121/288, 1331/10368, 14641/497664, 161051/29859840, 1771561/2149908480, 19487171/180592312320, 214358881/17336861982720, 2357947691/1872381094133760, 25937424601/224685731296051200, 25937424601/2696228775552614400, 285311670611/388256943679576473600, 3138428376721/60568083214013929881600, 34522712143931/10175437979954340220108800, 379749833583241/1831578836391781239619584000], [1, 23/12, 673/288, 25559/10368, 1241473/497664, 74649431/29859840, 5376530593/2149908480, 451648056983/180592312320, 43358427829249/17336861982720, 4682712563506583/1872381094133760, 561925533558214561/224685731296051200, 6743106428635999333/2696228775552614400, 971007326008895574563/388256943679576473600, 21639591837218019715507/8652583316287704268800, 25448160000602913897580163/10175437979954340220108800, 4580668800108904251398012581/1831578836391781239619584000], 4580668800108904251398012581/1831578836391781239619584000, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (11/12, 15, [1, 11/12, 121/288, 1331/10368, 14641/497664, 161051/29859840, 1771561/2149908480, 19487171/180592312320, 214358881/17336861982720, 2357947691/1872381094133760, 25937424601/224685731296051200, 25937424601/2696228775552614400, 285311670611/388256943679576473600, 3138428376721/60568083214013929881600, 34522712143931/10175437979954340220108800, 379749833583241/1831578836391781239619584000], [1, 23/12, 673/288, 25559/10368, 1241473/497664, 74649431/29859840, 5376530593/2149908480, 451648056983/180592312320, 43358427829249/17336861982720, 4682712563506583/1872381094133760, 561925533558214561/224685731296051200, 6743106428635999333/2696228775552614400, 971007326008895574563/388256943679576473600, 21639591837218019715507/8652583316287704268800, 25448160000602913897580163/10175437979954340220108800, 4580668800108904251398012581/1831578836391781239619584000], 4580668800108904251398012581/1831578836391781239619584000, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0012","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76aa6e8d35e7d76444d2360e854ac63cd5bb0bdd7b7f3a6d5c02090ccd64d204","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132012,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":16,\"x\":[12,13]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=12/13, cutoff=16, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(12/13, 16, [1, 12/13, 72/169, 288/2197, 864/28561, 10368/1856465, 20736/24134045, 248832/2196198095, 373248/28550575235, 497664/371157478055, 2985984/24125236073575, 35831808/3449908758521225, 35831808/44848813860775925, 429981696/7579449542471131325, 2579890176/689729908364872950575, 10319560704/44832444043716741787375, 7739670528/582821772568317643235875], [1, 25/13, 397/169, 5449/2197, 71701/28561, 4670933/1856465, 12148573/4826809, 5527849547/2196198095, 71862417359/28550575235, 934211923331/371157478055, 60723778002499/24125236073575, 248100008291119/98568821672035, 112885503808290953/44848813860775925, 19077650144031152753/7579449542471131325, 1736066163109414790699/689729908364872950575, 112844300602122280956139/44832444043716741787375, 293395181565519478420067/116564354513663528647175], 293395181565519478420067/116564354513663528647175, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (12/13, 16, [1, 12/13, 72/169, 288/2197, 864/28561, 10368/1856465, 20736/24134045, 248832/2196198095, 373248/28550575235, 497664/371157478055, 2985984/24125236073575, 35831808/3449908758521225, 35831808/44848813860775925, 429981696/7579449542471131325, 2579890176/689729908364872950575, 10319560704/44832444043716741787375, 7739670528/582821772568317643235875], [1, 25/13, 397/169, 5449/2197, 71701/28561, 4670933/1856465, 12148573/4826809, 5527849547/2196198095, 71862417359/28550575235, 934211923331/371157478055, 60723778002499/24125236073575, 248100008291119/98568821672035, 112885503808290953/44848813860775925, 19077650144031152753/7579449542471131325, 1736066163109414790699/689729908364872950575, 112844300602122280956139/44832444043716741787375, 293395181565519478420067/116564354513663528647175], 293395181565519478420067/116564354513663528647175, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0013","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92c4f30e01283e9cab5ffd5d06a53af800ac227bc0e3c36a573a55eee8657c83","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132013,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":17,\"x\":[13,14]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=13/14, cutoff=17, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(13/14, 17, [1, 13/14, 169/392, 2197/16464, 28561/921984, 371293/64538880, 4826809/5421265920, 62748517/531284060160, 815730721/59503814737920, 10604499373/7497480656977920, 137858491849/1049647291976908800, 1792160394037/161645682964443955200, 23298085122481/27156474738026584473600, 23298085122481/380190646332372182630400, 302875106592253/74517366681144947795558400, 3937376385699289/15648647003040439037067264000, 51185893014090757/3505296928681058344303067136000, 665416609183179841/834260669026091885944129978368000], [1, 27/14, 925/392, 41047/16464, 775731/307328, 163274803/64538880, 13719910261/5421265920, 29880310091/11806312448, 150597578589361/59503814737920, 18975305506758859/7497480656977920, 98390478103878967/38875825628774400, 409107609748089138823/161645682964443955200, 13746015692195412088949/5431294947605316894720, 106913455386330770149879/42243405148041353625600, 188595335301790353650978809/74517366681144947795558400, 3600456401216355603917386289/1422604273003676276097024000, 985724952510794599327272656317/389477436520117593811451904000, 422284569655624539435125442602171/166852133805218377188825995673600], 422284569655624539435125442602171/166852133805218377188825995673600, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (13/14, 17, [1, 13/14, 169/392, 2197/16464, 28561/921984, 371293/64538880, 4826809/5421265920, 62748517/531284060160, 815730721/59503814737920, 10604499373/7497480656977920, 137858491849/1049647291976908800, 1792160394037/161645682964443955200, 23298085122481/27156474738026584473600, 23298085122481/380190646332372182630400, 302875106592253/74517366681144947795558400, 3937376385699289/15648647003040439037067264000, 51185893014090757/3505296928681058344303067136000, 665416609183179841/834260669026091885944129978368000], [1, 27/14, 925/392, 41047/16464, 775731/307328, 163274803/64538880, 13719910261/5421265920, 29880310091/11806312448, 150597578589361/59503814737920, 18975305506758859/7497480656977920, 98390478103878967/38875825628774400, 409107609748089138823/161645682964443955200, 13746015692195412088949/5431294947605316894720, 106913455386330770149879/42243405148041353625600, 188595335301790353650978809/74517366681144947795558400, 3600456401216355603917386289/1422604273003676276097024000, 985724952510794599327272656317/389477436520117593811451904000, 422284569655624539435125442602171/166852133805218377188825995673600], 422284569655624539435125442602171/166852133805218377188825995673600, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0014","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f73adbe6523f8333ca11dea4a5e436aca69ce49f924cf10924141c25890fbc61","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132014,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":18,\"x\":[14,15]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=14/15, cutoff=18, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(14/15, 18, [1, 14/15, 98/225, 1372/10125, 4802/151875, 67228/11390625, 470596/512578125, 941192/7688671875, 1647086/115330078125, 23059204/15569560546875, 161414428/1167717041015625, 2259801992/192673311767578125, 7909306972/8670299029541015625, 110730297608/1690708310760498046875, 110730297608/25360624661407470703125, 1550224166512/5706140548816680908203125, 1356446145698/85592108232250213623046875, 18990246039772/21825987599223804473876953125, 132931722278404/2946508325895213603973388671875], [1, 29/15, 533/225, 25357/10125, 385157/151875, 28954003/11390625, 1303400731/512578125, 19551952157/7688671875, 293280929441/115330078125, 39592948533739/15569560546875, 2969471301444853/1167717041015625, 489962766998202737/192673311767578125, 22048324522828430137/8670299029541015625, 4299423282062274174323/1690708310760498046875, 64491349231044842912453/25360624661407470703125, 1116196428998972298420649/438933888370513916015625, 217658303654800954638172253/85592108232250213623046875, 55502867431974262422979964287/21825987599223804473876953125, 576375931024348120002616727473/226654486607324123382568359375], 576375931024348120002616727473/226654486607324123382568359375, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (14/15, 18, [1, 14/15, 98/225, 1372/10125, 4802/151875, 67228/11390625, 470596/512578125, 941192/7688671875, 1647086/115330078125, 23059204/15569560546875, 161414428/1167717041015625, 2259801992/192673311767578125, 7909306972/8670299029541015625, 110730297608/1690708310760498046875, 110730297608/25360624661407470703125, 1550224166512/5706140548816680908203125, 1356446145698/85592108232250213623046875, 18990246039772/21825987599223804473876953125, 132931722278404/2946508325895213603973388671875], [1, 29/15, 533/225, 25357/10125, 385157/151875, 28954003/11390625, 1303400731/512578125, 19551952157/7688671875, 293280929441/115330078125, 39592948533739/15569560546875, 2969471301444853/1167717041015625, 489962766998202737/192673311767578125, 22048324522828430137/8670299029541015625, 4299423282062274174323/1690708310760498046875, 64491349231044842912453/25360624661407470703125, 1116196428998972298420649/438933888370513916015625, 217658303654800954638172253/85592108232250213623046875, 55502867431974262422979964287/21825987599223804473876953125, 576375931024348120002616727473/226654486607324123382568359375], 576375931024348120002616727473/226654486607324123382568359375, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0015","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03f206cee7ac6d0ee63ea411f3e832798e7fa9f3226f7815b3ac7f74d90da768","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132015,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":19,\"x\":[15,16]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=15/16, cutoff=19, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(15/16, 19, [1, 15/16, 225/512, 1125/8192, 16875/524288, 50625/8388608, 253125/268435456, 3796875/30064771072, 56953125/3848290697216, 94921875/61572651155456, 284765625/1970324836974592, 4271484375/346777171307528192, 21357421875/22193738963681804288, 320361328125/4616297704445815291904, 4805419921875/1034050685795862625386496, 4805419921875/16544810972733802006183936, 72081298828125/4235471609019853313583087616, 1081219482421875/1152048277653400101294599831552, 1802032470703125/36865544884908803241427194609664, 27030487060546875/11207125645012276185393867161337856], [1, 31/16, 1217/512, 20597/8192, 1335083/524288, 21411953/8388608, 685435621/268435456, 76772586427/30064771072, 9826948015781/3848290697216, 157231263174371/61572651155456, 5031400706345497/1970324836974592, 885526528588291847/346777171307528192, 56673697851008100083/22193738963681804288, 11788129153330046145389/4616297704445815291904, 2640540930350735756489011/1034050685795862625386496, 42248654885616577523746051/16544810972733802006183936, 10815655650717915927377817181/4235471609019853313583087616, 2941858336995274213466248695107/1152048277653400101294599831552, 94139466783848776632952428946549/36865544884908803241427194609664, 2201415223253079086419078881561367/862086588077867398876451320102912], 2201415223253079086419078881561367/862086588077867398876451320102912, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (15/16, 19, [1, 15/16, 225/512, 1125/8192, 16875/524288, 50625/8388608, 253125/268435456, 3796875/30064771072, 56953125/3848290697216, 94921875/61572651155456, 284765625/1970324836974592, 4271484375/346777171307528192, 21357421875/22193738963681804288, 320361328125/4616297704445815291904, 4805419921875/1034050685795862625386496, 4805419921875/16544810972733802006183936, 72081298828125/4235471609019853313583087616, 1081219482421875/1152048277653400101294599831552, 1802032470703125/36865544884908803241427194609664, 27030487060546875/11207125645012276185393867161337856], [1, 31/16, 1217/512, 20597/8192, 1335083/524288, 21411953/8388608, 685435621/268435456, 76772586427/30064771072, 9826948015781/3848290697216, 157231263174371/61572651155456, 5031400706345497/1970324836974592, 885526528588291847/346777171307528192, 56673697851008100083/22193738963681804288, 11788129153330046145389/4616297704445815291904, 2640540930350735756489011/1034050685795862625386496, 42248654885616577523746051/16544810972733802006183936, 10815655650717915927377817181/4235471609019853313583087616, 2941858336995274213466248695107/1152048277653400101294599831552, 94139466783848776632952428946549/36865544884908803241427194609664, 2201415223253079086419078881561367/862086588077867398876451320102912], 2201415223253079086419078881561367/862086588077867398876451320102912, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0016","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e870e7e22d309cf7693265dc013420837bde305dbe9503eca1e3e0254c026b9d","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132016,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":20,\"x\":[16,17]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=16/17, cutoff=20, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(16/17, 20, [1, 16/17, 128/289, 2048/14739, 8192/250563, 131072/21297855, 1048576/1086190605, 16777216/129256681995, 33554432/2197363593915, 536870912/336196629868995, 4294967296/28576713538864575, 68719476736/5343845431767675525, 274877906944/272536117020151451775, 4398046511104/60230481861453470842275, 35184372088832/7167427341512963030230725, 562949953421312/1827693972085805572708834875, 562949953421312/31070797525458694736050192875, 9007199254740992/8979460484857562778718505740875, 72057594037927936/1373857454183207105143931378353875, 1152921504606846976/443755957701175894961489835208301625, 4611686018427387904/37719256404599951071726635992705638125], [1, 33/17, 689/289, 37187/14739, 213457/83521, 54562607/21297855, 2783741533/1086190605, 110427339881/43085560665, 5631827888363/2197363593915, 861670203790451/336196629868995, 24413990539051877/9525571179621525, 13696248761127579733/5343845431767675525, 63500789735671315757/24776010638195586525, 51456806617271671705457/20076827287151156947425, 18370079962401171170936981/7167427341512963030230725, 360336183877912430657103959/140591844006600428669910375, 2413160504152097276520405947/941539318953293779880308875, 23014311728098560733374366257531/8979460484857562778718505740875, 3521189694399079864263872075330179/1373857454183207105143931378353875, 379114757096967599103384061646164931/147918652567058631653829945069433875, 8788569369066067070543147430745404119/3429023309509086461066057817518694375], 8788569369066067070543147430745404119/3429023309509086461066057817518694375, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (16/17, 20, [1, 16/17, 128/289, 2048/14739, 8192/250563, 131072/21297855, 1048576/1086190605, 16777216/129256681995, 33554432/2197363593915, 536870912/336196629868995, 4294967296/28576713538864575, 68719476736/5343845431767675525, 274877906944/272536117020151451775, 4398046511104/60230481861453470842275, 35184372088832/7167427341512963030230725, 562949953421312/1827693972085805572708834875, 562949953421312/31070797525458694736050192875, 9007199254740992/8979460484857562778718505740875, 72057594037927936/1373857454183207105143931378353875, 1152921504606846976/443755957701175894961489835208301625, 4611686018427387904/37719256404599951071726635992705638125], [1, 33/17, 689/289, 37187/14739, 213457/83521, 54562607/21297855, 2783741533/1086190605, 110427339881/43085560665, 5631827888363/2197363593915, 861670203790451/336196629868995, 24413990539051877/9525571179621525, 13696248761127579733/5343845431767675525, 63500789735671315757/24776010638195586525, 51456806617271671705457/20076827287151156947425, 18370079962401171170936981/7167427341512963030230725, 360336183877912430657103959/140591844006600428669910375, 2413160504152097276520405947/941539318953293779880308875, 23014311728098560733374366257531/8979460484857562778718505740875, 3521189694399079864263872075330179/1373857454183207105143931378353875, 379114757096967599103384061646164931/147918652567058631653829945069433875, 8788569369066067070543147430745404119/3429023309509086461066057817518694375], 8788569369066067070543147430745404119/3429023309509086461066057817518694375, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0017","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88e545514f79658ad36a109aac71a0f1b967ebcdfb770d263255aa3f1953c018","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132017,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":21,\"x\":[17,18]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=17/18, cutoff=21, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(17/18, 21, [1, 17/18, 289/648, 4913/34992, 83521/2519424, 1419857/226748160, 24137569/24488801280, 410338673/3085588961280, 6975757441/444324810424320, 118587876497/71980619288739840, 2015993900449/12956511471973171200, 34271896307633/2565389271450687897600, 582622237229761/554124082633348585881600, 9904578032905937/129665035336203569096294400, 168377826559400929/32675588904723299412266188800, 2862423051509815793/8822409004275290841311870976000, 48661191875666868481/2540853793231283762297818841088000, 48661191875666868481/45735368278163107721360739139584000, 827240261886336764177/14818259322124846901720879481225216000, 14063084452067724991009/5067844688166697640388540782579023872000, 239072435685151324847153/1824424087740011150539874681728448593920000, 4064231406647572522401601/689632305165724214904072629693353568501760000], [1, 35/18, 1549/648, 88559/34992, 6459769/2519424, 582799067/226748160, 12593287361/4897760256, 7934181376103/3085588961280, 163218441988039/63474972917760, 185089831802312723/71980619288739840, 33316171740410190589/12956511471973171200, 1319320407774622808851/513077854290137579520, 1424866040979214870788841/554124082633348585881600, 333418653599040857797494731/129665035336203569096294400, 84021500707126673991528073141/32675588904723299412266188800, 462975616141368661240083460487/180049163352556955945140224000, 261340475799481728344477338450441/101634151729251350491912753643520, 117603214109766826416206677969566931/45735368278163107721360739139584000, 38103441371564452586091225548476449821/14818259322124846901720879481225216000, 13031376949075042798506283589646670829791/5067844688166697640388540782579023872000, 426481427424274127972848593450722983961083/165856735249091922776352243793495326720000, 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000], 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (17/18, 21, [1, 17/18, 289/648, 4913/34992, 83521/2519424, 1419857/226748160, 24137569/24488801280, 410338673/3085588961280, 6975757441/444324810424320, 118587876497/71980619288739840, 2015993900449/12956511471973171200, 34271896307633/2565389271450687897600, 582622237229761/554124082633348585881600, 9904578032905937/129665035336203569096294400, 168377826559400929/32675588904723299412266188800, 2862423051509815793/8822409004275290841311870976000, 48661191875666868481/2540853793231283762297818841088000, 48661191875666868481/45735368278163107721360739139584000, 827240261886336764177/14818259322124846901720879481225216000, 14063084452067724991009/5067844688166697640388540782579023872000, 239072435685151324847153/1824424087740011150539874681728448593920000, 4064231406647572522401601/689632305165724214904072629693353568501760000], [1, 35/18, 1549/648, 88559/34992, 6459769/2519424, 582799067/226748160, 12593287361/4897760256, 7934181376103/3085588961280, 163218441988039/63474972917760, 185089831802312723/71980619288739840, 33316171740410190589/12956511471973171200, 1319320407774622808851/513077854290137579520, 1424866040979214870788841/554124082633348585881600, 333418653599040857797494731/129665035336203569096294400, 84021500707126673991528073141/32675588904723299412266188800, 462975616141368661240083460487/180049163352556955945140224000, 261340475799481728344477338450441/101634151729251350491912753643520, 117603214109766826416206677969566931/45735368278163107721360739139584000, 38103441371564452586091225548476449821/14818259322124846901720879481225216000, 13031376949075042798506283589646670829791/5067844688166697640388540782579023872000, 426481427424274127972848593450722983961083/165856735249091922776352243793495326720000, 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000], 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0018","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"918c75f99c2cfd2dfc9e16dc659b49db13882dda9756240bdb80a50fe4feae51","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132018,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":22,\"x\":[18,19]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=18/19, cutoff=22, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(18/19, 22, [1, 18/19, 162/361, 972/6859, 4374/130321, 78732/12380495, 236196/235229405, 4251528/31285510865, 9565938/594424706435, 19131876/11294069422265, 172186884/1072936595115175, 3099363912/224243748379071575, 4649045868/4260631219202359925, 83682825624/1052375911142982901475, 753145430616/139965996182016725896175, 4518872583696/13296769637291588960136625, 5083731656658/252638623108540190242595875, 91507169819844/81602275264058481448358467625, 91507169819844/1550443230017111147518810884875, 1647129056757192/559710006036177124254290729439875, 7412080755407364/53172450573436826804157619296788125, 44472484532444184/7071935926267097964952963366472820625, 400252360791997656/1478034608589823474675169343592819510625], [1, 37/19, 865/361, 17407/6859, 335107/130321, 31913897/12380495, 606600239/235229405, 16136416663/6257102173, 1532969148923/594424706435, 29126432961413/11294069422265, 2767011303521119/1072936595115175, 578305365535277783/224243748379071575, 2197560389963864749/852126243840471985, 387712440241293684377/150339415877568985925, 360960281865397565585603/139965996182016725896175, 34291226777217287603215981/13296769637291588960136625, 50117946828241042168673869/19433740239118476172507375, 8417810349271369102937255831/3264091010562339257934338705, 3998459915903900415402366339569/1550443230017111147518810884875, 1443444029641308051607383305341601/559710006036177124254290729439875, 19589597545132037844301927823265637/7596064367633832400593945613826875, 18237915314517927233089567287992752231/7071935926267097964952963366472820625, 762344860146849358343223963110255442787/295606921717964694935033868718563902125], 762344860146849358343223963110255442787/295606921717964694935033868718563902125, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (18/19, 22, [1, 18/19, 162/361, 972/6859, 4374/130321, 78732/12380495, 236196/235229405, 4251528/31285510865, 9565938/594424706435, 19131876/11294069422265, 172186884/1072936595115175, 3099363912/224243748379071575, 4649045868/4260631219202359925, 83682825624/1052375911142982901475, 753145430616/139965996182016725896175, 4518872583696/13296769637291588960136625, 5083731656658/252638623108540190242595875, 91507169819844/81602275264058481448358467625, 91507169819844/1550443230017111147518810884875, 1647129056757192/559710006036177124254290729439875, 7412080755407364/53172450573436826804157619296788125, 44472484532444184/7071935926267097964952963366472820625, 400252360791997656/1478034608589823474675169343592819510625], [1, 37/19, 865/361, 17407/6859, 335107/130321, 31913897/12380495, 606600239/235229405, 16136416663/6257102173, 1532969148923/594424706435, 29126432961413/11294069422265, 2767011303521119/1072936595115175, 578305365535277783/224243748379071575, 2197560389963864749/852126243840471985, 387712440241293684377/150339415877568985925, 360960281865397565585603/139965996182016725896175, 34291226777217287603215981/13296769637291588960136625, 50117946828241042168673869/19433740239118476172507375, 8417810349271369102937255831/3264091010562339257934338705, 3998459915903900415402366339569/1550443230017111147518810884875, 1443444029641308051607383305341601/559710006036177124254290729439875, 19589597545132037844301927823265637/7596064367633832400593945613826875, 18237915314517927233089567287992752231/7071935926267097964952963366472820625, 762344860146849358343223963110255442787/295606921717964694935033868718563902125], 762344860146849358343223963110255442787/295606921717964694935033868718563902125, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0019","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d66d82bb0bb17d4e21252a3cb055e0a948786881fcadcb7b7aae1dc0c3b26652","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132019,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":23,\"x\":[19,20]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=19/20, cutoff=23, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(19/20, 23, [1, 19/20, 361/800, 6859/48000, 130321/3840000, 2476099/384000000, 47045881/46080000000, 893871739/6451200000000, 16983563041/1032192000000000, 322687697779/185794560000000000, 6131066257801/37158912000000000000, 116490258898219/8174960640000000000000, 2213314919066161/1961990553600000000000000, 42052983462257059/510117543936000000000000000, 799006685782884121/142832912302080000000000000000, 15181127029874798299/42849873690624000000000000000000, 288441413567621167681/13711959580999680000000000000000000, 5480386857784802185939/4662066257539891200000000000000000000, 104127350297911241532841/1678343852714360832000000000000000000000, 104127350297911241532841/33566877054287216640000000000000000000000, 1978419655660313589123979/13426750821714886656000000000000000000000000, 37589973457545958193355601/5639235345120252395520000000000000000000000000, 714209495693373205673756419/2481263551852911054028800000000000000000000000000, 13569980418174090907801371961/1141381233852339084853248000000000000000000000000000], [1, 39/20, 1921/800, 122119/48000, 3299947/1280000, 992460199/384000000, 119142269761/46080000000, 5560270546093/2150400000000, 2668946845687681/1032192000000000, 480410754911480359/185794560000000000, 32027385704454109867/12386304000000000000, 3019724954489995915777/1167851520000000000000, 5073137925756508057571521/1961990553600000000000000, 439671953579581692810284173/170039181312000000000000000, 28409572385203663742032429957/10987147100160000000000000000, 110797332302309469720956351630599/42849873690624000000000000000000, 11818382112246439584039866714319787/4570653193666560000000000000000000, 12054749754491373856107521833408368679/4662066257539891200000000000000000000, 619958558802413527475151165419750608183/239763407530622976000000000000000000000, 1701847024163488116679382616797575032911/658174059887984640000000000000000000000, 34717679292935157582237825038330844260508379/13426750821714886656000000000000000000000000, 1121648100233289706505959729965884657508221137/433787334240019415040000000000000000000000000, 2138609044444805707071601288300184538050899553353/827087850617637018009600000000000000000000000000, 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000], 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (19/20, 23, [1, 19/20, 361/800, 6859/48000, 130321/3840000, 2476099/384000000, 47045881/46080000000, 893871739/6451200000000, 16983563041/1032192000000000, 322687697779/185794560000000000, 6131066257801/37158912000000000000, 116490258898219/8174960640000000000000, 2213314919066161/1961990553600000000000000, 42052983462257059/510117543936000000000000000, 799006685782884121/142832912302080000000000000000, 15181127029874798299/42849873690624000000000000000000, 288441413567621167681/13711959580999680000000000000000000, 5480386857784802185939/4662066257539891200000000000000000000, 104127350297911241532841/1678343852714360832000000000000000000000, 104127350297911241532841/33566877054287216640000000000000000000000, 1978419655660313589123979/13426750821714886656000000000000000000000000, 37589973457545958193355601/5639235345120252395520000000000000000000000000, 714209495693373205673756419/2481263551852911054028800000000000000000000000000, 13569980418174090907801371961/1141381233852339084853248000000000000000000000000000], [1, 39/20, 1921/800, 122119/48000, 3299947/1280000, 992460199/384000000, 119142269761/46080000000, 5560270546093/2150400000000, 2668946845687681/1032192000000000, 480410754911480359/185794560000000000, 32027385704454109867/12386304000000000000, 3019724954489995915777/1167851520000000000000, 5073137925756508057571521/1961990553600000000000000, 439671953579581692810284173/170039181312000000000000000, 28409572385203663742032429957/10987147100160000000000000000, 110797332302309469720956351630599/42849873690624000000000000000000, 11818382112246439584039866714319787/4570653193666560000000000000000000, 12054749754491373856107521833408368679/4662066257539891200000000000000000000, 619958558802413527475151165419750608183/239763407530622976000000000000000000000, 1701847024163488116679382616797575032911/658174059887984640000000000000000000000, 34717679292935157582237825038330844260508379/13426750821714886656000000000000000000000000, 1121648100233289706505959729965884657508221137/433787334240019415040000000000000000000000000, 2138609044444805707071601288300184538050899553353/827087850617637018009600000000000000000000000000, 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000], 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0020","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"574901512034160b9e3e2b0b29298061d4f6deff759ecf36e9c5bcfbf9229b54","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132020,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":24,\"x\":[20,21]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=20/21, cutoff=24, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(20/21, 24, [1, 20/21, 200/441, 4000/27783, 20000/583443, 80000/12252303, 800000/771895089, 16000000/113468578083, 40000000/2382840139743, 800000000/450356786411427, 1600000000/9457492514639967, 32000000000/2184680770881832377, 160000000000/137634888565555439751, 3200000000000/37574324578396635052023, 32000000000000/5523425713024305352647381, 128000000000000/347975819920531237216785003, 160000000000000/7307492218331155981552485063, 3200000000000000/2608774721944222685414237167491, 32000000000000000/493058422447458087543290824655799, 640000000000000000/196730310556535776929773039037663801, 640000000000000000/4131336521687251315525233819790939821, 12800000000000000000/1821919406064077830146628114527804461061, 128000000000000000000/420863382800801978763871094455922830505091, 2560000000000000000000/203277013892787355742949738622210727133958953, 6400000000000000000000/12806451875245603411805833533199275809439414039], [1, 41/21, 1061/441, 70843/27783, 1507703/583443, 31741763/12252303, 2000531069/771895089, 294094067143/113468578083, 6176015410003/2382840139743, 1167267712490567/450356786411427, 24512623562301907/9457492514639967, 5662416074891740517/2184680770881832377, 356732212878179652571/137634888565555439751, 97387894118943045151883/37574324578396635052023, 14316020435516627637326801/5523425713024305352647381, 901909287437675541151588463/347975819920531237216785003, 18940095036191346364183357723/7307492218331155981552485063, 6761613927920313852013458707111/2608774721944222685414237167491, 1277945032376939350030543695643979/493058422447458087543290824655799, 509900067918398801302186934561947621/196730310556535776929773039037663801, 10707901426286374827985925625800900041/4131336521687251315525233819790939821, 4722184528992291299154593200978196918081/1821919406064077830146628114527804461061, 1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091, 526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953, 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039], 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (20/21, 24, [1, 20/21, 200/441, 4000/27783, 20000/583443, 80000/12252303, 800000/771895089, 16000000/113468578083, 40000000/2382840139743, 800000000/450356786411427, 1600000000/9457492514639967, 32000000000/2184680770881832377, 160000000000/137634888565555439751, 3200000000000/37574324578396635052023, 32000000000000/5523425713024305352647381, 128000000000000/347975819920531237216785003, 160000000000000/7307492218331155981552485063, 3200000000000000/2608774721944222685414237167491, 32000000000000000/493058422447458087543290824655799, 640000000000000000/196730310556535776929773039037663801, 640000000000000000/4131336521687251315525233819790939821, 12800000000000000000/1821919406064077830146628114527804461061, 128000000000000000000/420863382800801978763871094455922830505091, 2560000000000000000000/203277013892787355742949738622210727133958953, 6400000000000000000000/12806451875245603411805833533199275809439414039], [1, 41/21, 1061/441, 70843/27783, 1507703/583443, 31741763/12252303, 2000531069/771895089, 294094067143/113468578083, 6176015410003/2382840139743, 1167267712490567/450356786411427, 24512623562301907/9457492514639967, 5662416074891740517/2184680770881832377, 356732212878179652571/137634888565555439751, 97387894118943045151883/37574324578396635052023, 14316020435516627637326801/5523425713024305352647381, 901909287437675541151588463/347975819920531237216785003, 18940095036191346364183357723/7307492218331155981552485063, 6761613927920313852013458707111/2608774721944222685414237167491, 1277945032376939350030543695643979/493058422447458087543290824655799, 509900067918398801302186934561947621/196730310556535776929773039037663801, 10707901426286374827985925625800900041/4131336521687251315525233819790939821, 4722184528992291299154593200978196918081/1821919406064077830146628114527804461061, 1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091, 526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953, 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039], 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0021","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0c76c18691fd521f6189ad2fab02a12be94aa6645e0ad395b25930ef7e663da","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132021,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":25,\"x\":[21,22]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=21/22, cutoff=25, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(21/22, 25, [1, 21/22, 441/968, 3087/21296, 64827/1874048, 1361367/206145280, 9529569/9070392320, 28588707/199548631040, 600362847/35120559063040, 1400846643/772652299386880, 29417779503/169983505865113600, 617773369563/41136008419357491200, 4324413586941/3619968740903459225600, 90812685325761/1035311059898389338521600, 272438055977283/45553686635529130894950400, 1907066391840981/5010905529908204398444544000, 40048394228660601/1763838746527687948252479488000, 841016278801872621/659675691201355292646427328512000, 1962371317204369449/29025730412859632876442802454528000, 41209797661291758429/12132755312575326542353091425992704000, 865405750887126927009/5338412337533143678635360227436789760000, 865405750887126927009/117445071425729160929977925003609374720000, 18173520768629665467189/56843414570052913890109315701746937364480000, 381643936141222974810969/28762767772446774428395313745083950306426880000, 2671507552988560823676783/5062247127950632299397575219134775253931130880000, 56101658612759777297212443/2784235920372847764668666370524126389662121984000000], [1, 43/22, 2333/968, 54413/21296, 4853171/1874048, 535210177/206145280, 23558777357/9070392320, 518321690561/199548631040, 91225217901583/35120559063040, 2006956194681469/772652299386880, 441530392247702683/169983505865113600, 106850355541717418849/41136008419357491200, 9402831291995546445653/3619968740903459225600, 2689209749601538968782519/1035311059898389338521600, 118325228982740152682408119/45553686635529130894950400, 13015775188103323861456734071/5010905529908204398444544000, 4581552866212410047626999053593/1763838746527687948252479488000, 1713500771963442198828776447916403/659675691201355292646427328512000, 75394033966391458710837480912691181/29025730412859632876442802454528000, 31514706197951629782339864682796672087/12132755312575326542353091425992704000, 13866470727098717105094946211317662645289/5338412337533143678635360227436789760000, 305062355996171776312954222399875705123367/117445071425729160929977925003609374720000, 147650180302147139735488017162308470945176817/56843414570052913890109315701746937364480000, 74710991232886452706157318328064227521234280371/28762767772446774428395313745083950306426880000, 13149134456988015676283690697246857032298057022079/5062247127950632299397575219134775253931130880000, 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000], 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (21/22, 25, [1, 21/22, 441/968, 3087/21296, 64827/1874048, 1361367/206145280, 9529569/9070392320, 28588707/199548631040, 600362847/35120559063040, 1400846643/772652299386880, 29417779503/169983505865113600, 617773369563/41136008419357491200, 4324413586941/3619968740903459225600, 90812685325761/1035311059898389338521600, 272438055977283/45553686635529130894950400, 1907066391840981/5010905529908204398444544000, 40048394228660601/1763838746527687948252479488000, 841016278801872621/659675691201355292646427328512000, 1962371317204369449/29025730412859632876442802454528000, 41209797661291758429/12132755312575326542353091425992704000, 865405750887126927009/5338412337533143678635360227436789760000, 865405750887126927009/117445071425729160929977925003609374720000, 18173520768629665467189/56843414570052913890109315701746937364480000, 381643936141222974810969/28762767772446774428395313745083950306426880000, 2671507552988560823676783/5062247127950632299397575219134775253931130880000, 56101658612759777297212443/2784235920372847764668666370524126389662121984000000], [1, 43/22, 2333/968, 54413/21296, 4853171/1874048, 535210177/206145280, 23558777357/9070392320, 518321690561/199548631040, 91225217901583/35120559063040, 2006956194681469/772652299386880, 441530392247702683/169983505865113600, 106850355541717418849/41136008419357491200, 9402831291995546445653/3619968740903459225600, 2689209749601538968782519/1035311059898389338521600, 118325228982740152682408119/45553686635529130894950400, 13015775188103323861456734071/5010905529908204398444544000, 4581552866212410047626999053593/1763838746527687948252479488000, 1713500771963442198828776447916403/659675691201355292646427328512000, 75394033966391458710837480912691181/29025730412859632876442802454528000, 31514706197951629782339864682796672087/12132755312575326542353091425992704000, 13866470727098717105094946211317662645289/5338412337533143678635360227436789760000, 305062355996171776312954222399875705123367/117445071425729160929977925003609374720000, 147650180302147139735488017162308470945176817/56843414570052913890109315701746937364480000, 74710991232886452706157318328064227521234280371/28762767772446774428395313745083950306426880000, 13149134456988015676283690697246857032298057022079/5062247127950632299397575219134775253931130880000, 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000], 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0022","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e8e154bfa1ee56b939ad6f5f524d5317def42d24a96c107b2757d3a187901d4","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132022,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":26,\"x\":[22,23]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=22/23, cutoff=26, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(22/23, 26, [1, 22/23, 242/529, 5324/36501, 29282/839523, 644204/96545145, 7086244/6661615005, 155897368/1072520015805, 428717762/24667960363515, 9431790764/5106267795247605, 103749698404/587220796453474575, 207499396808/13506078318429915225, 1141246682444/931919403971664150525, 25107427013768/278643901787527581006975, 276181697151448/44861668187791940542122975, 6075997337331856/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625, 183798919454288644/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625, 44479338507937851848/12590729388204480196371154053302932125, 244636361793658185164/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125, 5381999959460480073608/16085097468959810607671022293486361392311875, 118403999108130561619376/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875], [1, 45/23, 1277/529, 93437/36501, 726111/279841, 251152499/96545145, 3467321735/1332323001, 310149988227/119168890645, 64201476280751/24667960363515, 13289715021906221/5106267795247605, 169813036807657091/65246755161497175, 7030259765336882929/2701215663685983045, 2425439620182471292949/931919403971664150525, 26859498017024679392797/10320144510649169666925, 116758237880282463017640007/44861668187791940542122975, 40281592068703525738423134271/15477275524788219487032426375, 6862789759853255159111636441/2636869163482437394087005975, 362252357473854257372627183826829/139187138794420457846882610390375, 5768172153622141022917072388409899/2216287517726541136484976950062125, 3640998444969709243963469638499387563/1398969932022720021819017117033659125, 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375], 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (22/23, 26, [1, 22/23, 242/529, 5324/36501, 29282/839523, 644204/96545145, 7086244/6661615005, 155897368/1072520015805, 428717762/24667960363515, 9431790764/5106267795247605, 103749698404/587220796453474575, 207499396808/13506078318429915225, 1141246682444/931919403971664150525, 25107427013768/278643901787527581006975, 276181697151448/44861668187791940542122975, 6075997337331856/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625, 183798919454288644/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625, 44479338507937851848/12590729388204480196371154053302932125, 244636361793658185164/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125, 5381999959460480073608/16085097468959810607671022293486361392311875, 118403999108130561619376/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875], [1, 45/23, 1277/529, 93437/36501, 726111/279841, 251152499/96545145, 3467321735/1332323001, 310149988227/119168890645, 64201476280751/24667960363515, 13289715021906221/5106267795247605, 169813036807657091/65246755161497175, 7030259765336882929/2701215663685983045, 2425439620182471292949/931919403971664150525, 26859498017024679392797/10320144510649169666925, 116758237880282463017640007/44861668187791940542122975, 40281592068703525738423134271/15477275524788219487032426375, 6862789759853255159111636441/2636869163482437394087005975, 362252357473854257372627183826829/139187138794420457846882610390375, 5768172153622141022917072388409899/2216287517726541136484976950062125, 3640998444969709243963469638499387563/1398969932022720021819017117033659125, 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375], 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0023","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a207cca6b6347172325bf0b0c5aee3f8ebeb65d0bed7aa9ac3fc44e3baac1a8d","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132023,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":27,\"x\":[23,24]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=23/24, cutoff=27, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(23/24, 27, [1, 23/24, 529/1152, 12167/82944, 279841/7962624, 6436343/955514880, 148035889/137594142720, 3404825447/23115815976960, 78310985281/4438236667576320, 1801152661463/958659120196485120, 41426511213649/230078188847156428800, 952809757913927/60740641855649297203200, 21914624432020321/17493304854426997594521600, 504036361936467383/5457911114581223249490739200, 11592836324538749809/1833858134499291011828888371200, 266635235464391245607/660188928419744764258399813632000, 6132610415680998648961/253512548513181989475225528434688000, 141050039560662968926103/103433119793378251705892015601352704000, 3244150909895248285300369/44683107750739404736945350739784368128000, 74615470927590710561908487/20375497134337168560047079937341671866368000, 1716155831334586342923895201/9780238624481840908822598369924002495856640000, 39471584120695485887249589623/4929240266738847818046589578441697257911746560000, 907846434775996175406740561329/2602638860838111647928599297417216152177402183680000, 907846434775996175406740561329/62463332660114679550286383138013187652257652408320000, 20880467999847912034355032910567/35978879612226055420964956687495596087700407787192320000, 480250763996501976790165756943041/21587327767335633252578974012497357652620244672315392000000, 11045767571919545466173812409689943/13470492526817435149609279783798351175235032675524804608000000, 254052654154149545721997685422868689/8728879157377697976946813299901331561552301173740073385984000000], [1, 47/24, 2785/1152, 212687/82944, 20697793/7962624, 2490171503/955514880, 358732732321/137594142720, 12054100771075/4623163195392, 11572015051217281/4438236667576320, 2499557052215594159/958659120196485120, 599893733958253811809/230078188847156428800, 158371946717788764231503/60740641855649297203200, 9122224135327557706138637/3498660970885399518904320, 14230669651615026383512741103/5457911114581223249490739200, 4781505002954241701184819760417/1833858134499291011828888371200, 1721341801063793647661999504995727/660188928419744764258399813632000, 660995251608502893312623490917008129/253512548513181989475225528434688000, 53937212531253864304317988991421648547/20686623958675650341178403120270540800, 10591307187955304558316160556065367832899/4062100704612673157904122794525851648000, 53125996854783807739129332276814595611729871/20375497134337168560047079937341671866368000, 1500028146488013395088131489659152484503190193/575308154381284759342505786466117793873920000, 12852241159109298769154582187520313973110583163247/4929240266738847818046589578441697257911746560000, 1357196666401941950022905448289100354795558930151149/520527772167622329585719859483443230435480436736000, 162863599968233034002749561641126818571642478358699209/62463332660114679550286383138013187652257652408320000, 93809433581702227585583768385757047345178101889643654951/35978879612226055420964956687495596087700407787192320000, 56285660149021336551350261511704992403608837923951949913641/21587327767335633252578974012497357652620244672315392000000, 35122251932989314008042563194349682831771460330719829155801927/13470492526817435149609279783798351175235032675524804608000000, 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000], 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (23/24, 27, [1, 23/24, 529/1152, 12167/82944, 279841/7962624, 6436343/955514880, 148035889/137594142720, 3404825447/23115815976960, 78310985281/4438236667576320, 1801152661463/958659120196485120, 41426511213649/230078188847156428800, 952809757913927/60740641855649297203200, 21914624432020321/17493304854426997594521600, 504036361936467383/5457911114581223249490739200, 11592836324538749809/1833858134499291011828888371200, 266635235464391245607/660188928419744764258399813632000, 6132610415680998648961/253512548513181989475225528434688000, 141050039560662968926103/103433119793378251705892015601352704000, 3244150909895248285300369/44683107750739404736945350739784368128000, 74615470927590710561908487/20375497134337168560047079937341671866368000, 1716155831334586342923895201/9780238624481840908822598369924002495856640000, 39471584120695485887249589623/4929240266738847818046589578441697257911746560000, 907846434775996175406740561329/2602638860838111647928599297417216152177402183680000, 907846434775996175406740561329/62463332660114679550286383138013187652257652408320000, 20880467999847912034355032910567/35978879612226055420964956687495596087700407787192320000, 480250763996501976790165756943041/21587327767335633252578974012497357652620244672315392000000, 11045767571919545466173812409689943/13470492526817435149609279783798351175235032675524804608000000, 254052654154149545721997685422868689/8728879157377697976946813299901331561552301173740073385984000000], [1, 47/24, 2785/1152, 212687/82944, 20697793/7962624, 2490171503/955514880, 358732732321/137594142720, 12054100771075/4623163195392, 11572015051217281/4438236667576320, 2499557052215594159/958659120196485120, 599893733958253811809/230078188847156428800, 158371946717788764231503/60740641855649297203200, 9122224135327557706138637/3498660970885399518904320, 14230669651615026383512741103/5457911114581223249490739200, 4781505002954241701184819760417/1833858134499291011828888371200, 1721341801063793647661999504995727/660188928419744764258399813632000, 660995251608502893312623490917008129/253512548513181989475225528434688000, 53937212531253864304317988991421648547/20686623958675650341178403120270540800, 10591307187955304558316160556065367832899/4062100704612673157904122794525851648000, 53125996854783807739129332276814595611729871/20375497134337168560047079937341671866368000, 1500028146488013395088131489659152484503190193/575308154381284759342505786466117793873920000, 12852241159109298769154582187520313973110583163247/4929240266738847818046589578441697257911746560000, 1357196666401941950022905448289100354795558930151149/520527772167622329585719859483443230435480436736000, 162863599968233034002749561641126818571642478358699209/62463332660114679550286383138013187652257652408320000, 93809433581702227585583768385757047345178101889643654951/35978879612226055420964956687495596087700407787192320000, 56285660149021336551350261511704992403608837923951949913641/21587327767335633252578974012497357652620244672315392000000, 35122251932989314008042563194349682831771460330719829155801927/13470492526817435149609279783798351175235032675524804608000000, 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000], 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0024","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e01665059f4da2cd72a8b5503ac3efbb77e855addf77cac9349aace66aa21fe","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132024,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":28,\"x\":[24,25]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=24/25, cutoff=28, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(24/25, 28, [1, 24/25, 288/625, 2304/15625, 13824/390625, 331776/48828125, 1327104/1220703125, 31850496/213623046875, 95551488/5340576171875, 254803968/133514404296875, 3057647616/16689300537109375, 73383542784/4589557647705078125, 146767085568/114738941192626953125, 3522410053632/37290155887603759765625, 42268920643584/6525777280330657958984375, 338151365148672/815722160041332244873046875, 507227047723008/20393054001033306121826171875, 12173449145352192/8667047950439155101776123046875, 16231265527136256/216676198760978877544403076171875, 389550372651270144/102921194411464966833591461181640625, 2337302235907620864/12865149301433120854198932647705078125, 18698417887260966912/2251401127750796149484813213348388671875, 224381014647131602944/619135310131468941108323633670806884765625, 5385144351531158470656/356002803325594641137286089360713958740234375, 5385144351531158470656/8900070083139866028432152234017848968505859375, 129243464436747803295744/5562543801962416267770095146261155605316162109375, 1550921573240973639548928/1807826735637785287025280922534875571727752685546875, 1378596953991976568487936/45195668390944632175632023063371889293193817138671875, 8271581723951859410927616/7909241968415310630735604036090080626308917999267578125], [1, 49/25, 1513/625, 40129/15625, 1017049/390625, 127462901/48828125, 3187899629/1220703125, 557914285571/213623046875, 1992564670109/762939453125, 348699072073043/133514404296875, 43587387066777991/16689300537109375, 11986531516747490309/4589557647705078125, 299663288065454343293/114738941192626953125, 97390568624795071623857/37290155887603759765625, 17043349509381406454818559/6525777280330657958984375, 27667775177571609846980111/10593794286251068115234375, 53260467216825856182484436683/20393054001033306121826171875, 22635698567151001051005030942467/8667047950439155101776123046875, 565892464178775042506391300697931/216676198760978877544403076171875, 15811701204995185034122720028399257/6054187906556762754917144775390625, 33599865060614768199848082296256041989/12865149301433120854198932647705078125, 452305875815968033460931755364005254999/173184702134676626883447170257568359375, 230999072291726531374722200920137988317767/88447901447352705872617661952972412109375, 929771265974199288783262243847906934137482831/356002803325594641137286089360713958740234375, 23244281649354982219581561481342024884595541431/8900070083139866028432152234017848968505859375, 44977325172900507390831195216972848265077451053/17221497838892929621579241938889026641845703125, 39020617438225047631012435697955754709648587304443/14940716823452770967151082004420459270477294921875, 118037367750630769083812617987694754950678953164428011/45195668390944632175632023063371889293193817138671875, 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125], 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (24/25, 28, [1, 24/25, 288/625, 2304/15625, 13824/390625, 331776/48828125, 1327104/1220703125, 31850496/213623046875, 95551488/5340576171875, 254803968/133514404296875, 3057647616/16689300537109375, 73383542784/4589557647705078125, 146767085568/114738941192626953125, 3522410053632/37290155887603759765625, 42268920643584/6525777280330657958984375, 338151365148672/815722160041332244873046875, 507227047723008/20393054001033306121826171875, 12173449145352192/8667047950439155101776123046875, 16231265527136256/216676198760978877544403076171875, 389550372651270144/102921194411464966833591461181640625, 2337302235907620864/12865149301433120854198932647705078125, 18698417887260966912/2251401127750796149484813213348388671875, 224381014647131602944/619135310131468941108323633670806884765625, 5385144351531158470656/356002803325594641137286089360713958740234375, 5385144351531158470656/8900070083139866028432152234017848968505859375, 129243464436747803295744/5562543801962416267770095146261155605316162109375, 1550921573240973639548928/1807826735637785287025280922534875571727752685546875, 1378596953991976568487936/45195668390944632175632023063371889293193817138671875, 8271581723951859410927616/7909241968415310630735604036090080626308917999267578125], [1, 49/25, 1513/625, 40129/15625, 1017049/390625, 127462901/48828125, 3187899629/1220703125, 557914285571/213623046875, 1992564670109/762939453125, 348699072073043/133514404296875, 43587387066777991/16689300537109375, 11986531516747490309/4589557647705078125, 299663288065454343293/114738941192626953125, 97390568624795071623857/37290155887603759765625, 17043349509381406454818559/6525777280330657958984375, 27667775177571609846980111/10593794286251068115234375, 53260467216825856182484436683/20393054001033306121826171875, 22635698567151001051005030942467/8667047950439155101776123046875, 565892464178775042506391300697931/216676198760978877544403076171875, 15811701204995185034122720028399257/6054187906556762754917144775390625, 33599865060614768199848082296256041989/12865149301433120854198932647705078125, 452305875815968033460931755364005254999/173184702134676626883447170257568359375, 230999072291726531374722200920137988317767/88447901447352705872617661952972412109375, 929771265974199288783262243847906934137482831/356002803325594641137286089360713958740234375, 23244281649354982219581561481342024884595541431/8900070083139866028432152234017848968505859375, 44977325172900507390831195216972848265077451053/17221497838892929621579241938889026641845703125, 39020617438225047631012435697955754709648587304443/14940716823452770967151082004420459270477294921875, 118037367750630769083812617987694754950678953164428011/45195668390944632175632023063371889293193817138671875, 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125], 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0025","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72f7c5493409c47e2908685d97dc58263fb919b8fb2e60f1ee539ef64c918129","family_id":"qf-mod44-exp-finite-partial-sum","seed":202611132025,"semantic_template":"qf-mod44-exp-finite-partial-sum/{\"cutoff\":29,\"x\":[25,26]}"},"fields":{"title":"Exact Finite Exponential Sums Are Not General Infinite-Value Certificates","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=25/26, cutoff=29, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).","answer":{"kind":"expression","value":"(25/26, 29, [1, 25/26, 625/1352, 15625/105456, 390625/10967424, 1953125/285153024, 48828125/44483871744, 1220703125/8096064657408, 30517578125/1683981448740864, 762939453125/394051659005362176, 3814697265625/20490686268278833152, 95367431640625/5860336272727746281472, 2384185791015625/1828424917091056839819264, 59604644775390625/618007621976777211858911232, 1490116119384765625/224954774399546905116643688448, 7450580596923828125/17546472403164658599098207698944, 186264514923095703125/7299332519716497977224854402760704, 4656612873077392578125/3226304973714692105933385646020231168, 116415321826934814453125/1509910727698475905576824482337468186624, 2910383045673370361328125/745895899483047097354951294274709284192256, 14551915228366851806640625/77573173546236898124914934604569765555994624, 363797880709171295166015625/42354952756245346376203554294095091993573064704, 9094947017729282379150390625/24227032976572338127188433056222392620323793010688, 227373675443232059478759765625/14487765719990258200058682967620990786953628220391424, 5684341886080801486968994140625/9040365809273921116836618171795498251059064009524248576, 5684341886080801486968994140625/235049511041121949037752072466682954527535664247630462976, 142108547152020037174224853515625/158893469463798437549520400987477677260614109031398192971776, 3552713678800500929355621337890625/111543215563586503159763321493209329436951104540041531466186752, 88817841970012523233890533447265625/81203460930290974300307698047056391830100404105150234907383955456, 2220446049250313080847263336181640625/61227409541439394622432004327480519439895704695283277120167502413824], [1, 51/26, 3277/1352, 271231/105456, 9532883/3655808, 745517999/285153024, 116349635969/44483871744, 7058951483161/2698688219136, 4404816243070589/1683981448740864, 1030727763817970951/394051659005362176, 17865949177743918359/6830228756092944384, 15328984489871713592647/5860336272727746281472, 4782643163224160431921489/1828424917091056839819264, 76977780439493850988802567/29428934379846533898043392, 588418153680981113077791587773/224954774399546905116643688448, 45896615987123977400664667674419/17546472403164658599098207698944, 6364330750214586954397141616087143/2433110839905499325741618134253568, 8439102574784546958143482860324129743/3226304973714692105933385646020231168, 232323529705833417225086576798029833697/88818278099910347386872028372792246272, 650351000823196346922220039207741634905177/248631966494349032451650431424903094730752, 28987073179548180036326366780168891699579407/11081881935176699732130704943509966507999232, 110788593692233144099203171714514675370958509179/42354952756245346376203554294095091993573064704, 21123691863985786141584436389240041198189139213671/8075677658857446042396144352074130873441264336896, 37895903203990500338002706255972077141610794509091399/14487765719990258200058682967620990786953628220391424, 23647043599290072210913694388068462217166622742667173601/9040365809273921116836618171795498251059064009524248576, 10786370764588453990943088767967050942581292601374397443/4123675632300385070837755657310227272412906390309306368, 415620438301122309179019096549414954071562403690383135789301/158893469463798437549520400987477677260614109031398192971776, 5954398932395670633544314403698816559939536904489889442081223/2276392154358908227750271867208353661978593970204929213595648, 70801772905472787613264261136277667389377724326732408903789550827/27067820310096991433435899349018797276700134701716744969127985152, 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824], 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824, No, finite_partial_sum_is_not_the_infinite_exponential_value)","explanation":"Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (25/26, 29, [1, 25/26, 625/1352, 15625/105456, 390625/10967424, 1953125/285153024, 48828125/44483871744, 1220703125/8096064657408, 30517578125/1683981448740864, 762939453125/394051659005362176, 3814697265625/20490686268278833152, 95367431640625/5860336272727746281472, 2384185791015625/1828424917091056839819264, 59604644775390625/618007621976777211858911232, 1490116119384765625/224954774399546905116643688448, 7450580596923828125/17546472403164658599098207698944, 186264514923095703125/7299332519716497977224854402760704, 4656612873077392578125/3226304973714692105933385646020231168, 116415321826934814453125/1509910727698475905576824482337468186624, 2910383045673370361328125/745895899483047097354951294274709284192256, 14551915228366851806640625/77573173546236898124914934604569765555994624, 363797880709171295166015625/42354952756245346376203554294095091993573064704, 9094947017729282379150390625/24227032976572338127188433056222392620323793010688, 227373675443232059478759765625/14487765719990258200058682967620990786953628220391424, 5684341886080801486968994140625/9040365809273921116836618171795498251059064009524248576, 5684341886080801486968994140625/235049511041121949037752072466682954527535664247630462976, 142108547152020037174224853515625/158893469463798437549520400987477677260614109031398192971776, 3552713678800500929355621337890625/111543215563586503159763321493209329436951104540041531466186752, 88817841970012523233890533447265625/81203460930290974300307698047056391830100404105150234907383955456, 2220446049250313080847263336181640625/61227409541439394622432004327480519439895704695283277120167502413824], [1, 51/26, 3277/1352, 271231/105456, 9532883/3655808, 745517999/285153024, 116349635969/44483871744, 7058951483161/2698688219136, 4404816243070589/1683981448740864, 1030727763817970951/394051659005362176, 17865949177743918359/6830228756092944384, 15328984489871713592647/5860336272727746281472, 4782643163224160431921489/1828424917091056839819264, 76977780439493850988802567/29428934379846533898043392, 588418153680981113077791587773/224954774399546905116643688448, 45896615987123977400664667674419/17546472403164658599098207698944, 6364330750214586954397141616087143/2433110839905499325741618134253568, 8439102574784546958143482860324129743/3226304973714692105933385646020231168, 232323529705833417225086576798029833697/88818278099910347386872028372792246272, 650351000823196346922220039207741634905177/248631966494349032451650431424903094730752, 28987073179548180036326366780168891699579407/11081881935176699732130704943509966507999232, 110788593692233144099203171714514675370958509179/42354952756245346376203554294095091993573064704, 21123691863985786141584436389240041198189139213671/8075677658857446042396144352074130873441264336896, 37895903203990500338002706255972077141610794509091399/14487765719990258200058682967620990786953628220391424, 23647043599290072210913694388068462217166622742667173601/9040365809273921116836618171795498251059064009524248576, 10786370764588453990943088767967050942581292601374397443/4123675632300385070837755657310227272412906390309306368, 415620438301122309179019096549414954071562403690383135789301/158893469463798437549520400987477677260614109031398192971776, 5954398932395670633544314403698816559939536904489889442081223/2276392154358908227750271867208353661978593970204929213595648, 70801772905472787613264261136277667389377724326732408903789550827/27067820310096991433435899349018797276700134701716744969127985152, 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824], 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824, No, finite_partial_sum_is_not_the_infinite_exponential_value)."},"skills":["Exact Finite Exponential Sums Are Not General Infinite-Value Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0026","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cdc4b0fa1f9aa1c0b0ee6720142b3feeeb9cf55618b42787a6cffaf05df0937","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132026,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":2,\"start_index\":2}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/2, N=2, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(2, 2, epsilon=1/2, |a_(n+1)/a_n|=1/(n+1), 1/3, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (2, 2, epsilon=1/2, |a_(n+1)/a_n|=1/(n+1), 1/3, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0027","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e4c8b139cf181586cb9607e92fc9f3cc261c190046b51b15f8b869e18a99eac","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132027,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":3,\"start_index\":4}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/3, N=4, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(3, 4, epsilon=1/3, |a_(n+1)/a_n|=1/(n+1), 1/5, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (3, 4, epsilon=1/3, |a_(n+1)/a_n|=1/(n+1), 1/5, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0028","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cf541fb54ad424d980e44412d4cfd598db92c2924c439eea0e1a7b525967864","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132028,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":4,\"start_index\":6}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/4, N=6, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(4, 6, epsilon=1/4, |a_(n+1)/a_n|=1/(n+1), 1/7, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (4, 6, epsilon=1/4, |a_(n+1)/a_n|=1/(n+1), 1/7, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0029","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6d1a92af86e90ca72be537b49b5b9654e3552a0a7bd5b69702fcae2e169e7c3","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132029,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":5,\"start_index\":5}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/5, N=5, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(5, 5, epsilon=1/5, |a_(n+1)/a_n|=1/(n+1), 1/6, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (5, 5, epsilon=1/5, |a_(n+1)/a_n|=1/(n+1), 1/6, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0030","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bf0f2837b4b5feb6690c17de8cd4eebcaf53800527d96270810e0a54cd2fe4d","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132030,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":6,\"start_index\":7}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/6, N=7, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(6, 7, epsilon=1/6, |a_(n+1)/a_n|=1/(n+1), 1/8, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (6, 7, epsilon=1/6, |a_(n+1)/a_n|=1/(n+1), 1/8, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0031","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"597bee0fdddd79585707f4355e0ba3e4166e11644012d9f9bdf59786cd8c578a","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132031,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":7,\"start_index\":9}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/7, N=9, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(7, 9, epsilon=1/7, |a_(n+1)/a_n|=1/(n+1), 1/10, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (7, 9, epsilon=1/7, |a_(n+1)/a_n|=1/(n+1), 1/10, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0032","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5843a45b8466f9a297f7391e6bb79f5a4d8c60937822469e579901288e320fba","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132032,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":8,\"start_index\":8}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/8, N=8, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(8, 8, epsilon=1/8, |a_(n+1)/a_n|=1/(n+1), 1/9, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (8, 8, epsilon=1/8, |a_(n+1)/a_n|=1/(n+1), 1/9, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0033","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701986ffc2f9023eb5420b2313d4cdb41d9f923a4f278bd1bd8c58ccf120cf47","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132033,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":9,\"start_index\":10}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/9, N=10, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(9, 10, epsilon=1/9, |a_(n+1)/a_n|=1/(n+1), 1/11, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (9, 10, epsilon=1/9, |a_(n+1)/a_n|=1/(n+1), 1/11, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0034","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6a7cb705fef84927cc5325b044333fd5d9133daa047daa70f77e7ed449475ee","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132034,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":10,\"start_index\":12}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/10, N=12, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(10, 12, epsilon=1/10, |a_(n+1)/a_n|=1/(n+1), 1/13, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (10, 12, epsilon=1/10, |a_(n+1)/a_n|=1/(n+1), 1/13, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0035","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75744ec1e473d91b78036effb6777990cb20c52ee51234fd8120a35c6810334b","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132035,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":11,\"start_index\":11}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/11, N=11, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(11, 11, epsilon=1/11, |a_(n+1)/a_n|=1/(n+1), 1/12, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (11, 11, epsilon=1/11, |a_(n+1)/a_n|=1/(n+1), 1/12, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0036","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e084691082d02597e0dfb134bd1b9749cccb7ac344275479defc32e5f44b1d4c","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132036,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":12,\"start_index\":13}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/12, N=13, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(12, 13, epsilon=1/12, |a_(n+1)/a_n|=1/(n+1), 1/14, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (12, 13, epsilon=1/12, |a_(n+1)/a_n|=1/(n+1), 1/14, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0037","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e16a5da3f3affd1e080682fda9f1a7421950a102b536e27e46ba235044d2352","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132037,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":13,\"start_index\":15}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/13, N=15, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(13, 15, epsilon=1/13, |a_(n+1)/a_n|=1/(n+1), 1/16, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (13, 15, epsilon=1/13, |a_(n+1)/a_n|=1/(n+1), 1/16, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0038","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac7ec3a5e9418283e16bd7bbd324b6d3f91e86c3fd8babe005789b8342693a2e","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132038,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":14,\"start_index\":14}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/14, N=14, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(14, 14, epsilon=1/14, |a_(n+1)/a_n|=1/(n+1), 1/15, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (14, 14, epsilon=1/14, |a_(n+1)/a_n|=1/(n+1), 1/15, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0039","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cf744606dbdd7d3b52ac35960bb19651964c09a58a26586db3d8a74edf2dd0d","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132039,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":15,\"start_index\":16}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/15, N=16, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(15, 16, epsilon=1/15, |a_(n+1)/a_n|=1/(n+1), 1/17, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (15, 16, epsilon=1/15, |a_(n+1)/a_n|=1/(n+1), 1/17, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0040","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"065dcf48c226bf835b88b5fafb161444e9981571b23dbdeb35c08dc3defa6f0a","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132040,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":16,\"start_index\":18}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/16, N=18, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(16, 18, epsilon=1/16, |a_(n+1)/a_n|=1/(n+1), 1/19, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (16, 18, epsilon=1/16, |a_(n+1)/a_n|=1/(n+1), 1/19, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0041","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94bcf991f10c17d9c3615becc267e0d59ae2565cd9665e176f5051926709245d","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132041,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":17,\"start_index\":17}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/17, N=17, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(17, 17, epsilon=1/17, |a_(n+1)/a_n|=1/(n+1), 1/18, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (17, 17, epsilon=1/17, |a_(n+1)/a_n|=1/(n+1), 1/18, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0042","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e05745f697b721cefad1d4bf5c1f145cc8a6f233cf6b9e76370523fa9f95e21","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132042,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":18,\"start_index\":19}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/18, N=19, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(18, 19, epsilon=1/18, |a_(n+1)/a_n|=1/(n+1), 1/20, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (18, 19, epsilon=1/18, |a_(n+1)/a_n|=1/(n+1), 1/20, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0043","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d04a883814210c43dc99f99a49b13920bad5b58985cafb8096b0a929e0f3201","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132043,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":19,\"start_index\":21}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/19, N=21, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(19, 21, epsilon=1/19, |a_(n+1)/a_n|=1/(n+1), 1/22, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (19, 21, epsilon=1/19, |a_(n+1)/a_n|=1/(n+1), 1/22, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0044","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a167d80ccfa4295e671d5537f36174ddc3438e7050b1ae4a456931edc77c3f29","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132044,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":20,\"start_index\":20}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/20, N=20, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(20, 20, epsilon=1/20, |a_(n+1)/a_n|=1/(n+1), 1/21, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (20, 20, epsilon=1/20, |a_(n+1)/a_n|=1/(n+1), 1/21, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0045","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4d2a6c7c67a0a4215b8ab9ab2b04434c260c5ad864edf0409ceb99697f21ac5","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132045,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":21,\"start_index\":22}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/21, N=22, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(21, 22, epsilon=1/21, |a_(n+1)/a_n|=1/(n+1), 1/23, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (21, 22, epsilon=1/21, |a_(n+1)/a_n|=1/(n+1), 1/23, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0046","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d9343c99ef1df4227355953437c1af0372a9e54a5b6e3455a58a0eeea871553","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132046,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":22,\"start_index\":24}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/22, N=24, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(22, 24, epsilon=1/22, |a_(n+1)/a_n|=1/(n+1), 1/25, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (22, 24, epsilon=1/22, |a_(n+1)/a_n|=1/(n+1), 1/25, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0047","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd6818f3c156a44457eb95445e9aacfa033ffc85b4aabb191671dc69174cd5c4","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132047,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":23,\"start_index\":23}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/23, N=23, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(23, 23, epsilon=1/23, |a_(n+1)/a_n|=1/(n+1), 1/24, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (23, 23, epsilon=1/23, |a_(n+1)/a_n|=1/(n+1), 1/24, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0048","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ad45fc4a313166a1b9942ee235ba70aada371df99377c49ab72fde346fce4ff","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132048,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":24,\"start_index\":25}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/24, N=25, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(24, 25, epsilon=1/24, |a_(n+1)/a_n|=1/(n+1), 1/26, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (24, 25, epsilon=1/24, |a_(n+1)/a_n|=1/(n+1), 1/26, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0049","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f90e9d3615637f10c414f936a2bd725809f1f9db4e4c9ae41fe348cc25478cf0","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132049,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":25,\"start_index\":27}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/25, N=27, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(25, 27, epsilon=1/25, |a_(n+1)/a_n|=1/(n+1), 1/28, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (25, 27, epsilon=1/25, |a_(n+1)/a_n|=1/(n+1), 1/28, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0050","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e44253f5a491fcb1914edde090610a5d369b87deb49bad156c4829b0794c2bc7","family_id":"qf-mod44-exp-infinite-radius-certificate","seed":202611132050,"semantic_template":"qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":26,\"start_index\":26}"},"fields":{"title":"The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/26, N=26, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)","explanation":"For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)."},"skills":["The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0051","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21fbeff031b027dc64e4e24817cc70b316d6df54c7fbc4d045893acf57ec3ca1","family_id":"qf-mod44-exp-addition-convolution","seed":202611132051,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":0,\"x\":[1,2],\"y\":[-2,3]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=1/2, y=-2/3, declared degree=0. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(1/2, -2/3, 0, [[0, 1]], 1, 1, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (1/2, -2/3, 0, [[0, 1]], 1, 1, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0052","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b383722a7c0a5d7eacced71dd34b0f88ffd8a51f0e316ae2f57055c5f442d0a1","family_id":"qf-mod44-exp-addition-convolution","seed":202611132052,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":1,\"x\":[2,3],\"y\":[-3,4]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=2/3, y=-3/4, declared degree=1. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(2/3, -3/4, 1, [[0, -3/4], [1, 2/3]], -1/12, -1/12, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (2/3, -3/4, 1, [[0, -3/4], [1, 2/3]], -1/12, -1/12, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0053","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df239b85fe6881a14451297cd3ab491261b2b3fddcf73235659efbc5ba265989","family_id":"qf-mod44-exp-addition-convolution","seed":202611132053,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":2,\"x\":[3,4],\"y\":[-4,5]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=3/4, y=-4/5, declared degree=2. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(3/4, -4/5, 2, [[0, 8/25], [1, -3/5], [2, 9/32]], 1/800, 1/800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (3/4, -4/5, 2, [[0, 8/25], [1, -3/5], [2, 9/32]], 1/800, 1/800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0054","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebc971b10d99f57a27a181114cf03b72ba75197accd6e1a852f15f81a8da41c6","family_id":"qf-mod44-exp-addition-convolution","seed":202611132054,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":3,\"x\":[4,5],\"y\":[-5,6]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=4/5, y=-5/6, declared degree=3. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(4/5, -5/6, 3, [[0, -125/1296], [1, 5/18], [2, -4/15], [3, 32/375]], -1/162000, -1/162000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (4/5, -5/6, 3, [[0, -125/1296], [1, 5/18], [2, -4/15], [3, 32/375]], -1/162000, -1/162000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0055","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df343cc1eb9f164e8397b0a2195fa02e015d889c311025a68f0d254d7e2ef9de","family_id":"qf-mod44-exp-addition-convolution","seed":202611132055,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":4,\"x\":[5,6],\"y\":[-6,7]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=5/6, y=-6/7, declared degree=4. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(5/6, -6/7, 4, [[0, 54/2401], [1, -30/343], [2, 25/196], [3, -125/1512], [4, 625/31104]], 1/74680704, 1/74680704, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (5/6, -6/7, 4, [[0, 54/2401], [1, -30/343], [2, 25/196], [3, -125/1512], [4, 625/31104]], 1/74680704, 1/74680704, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0056","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"308fb9fdacb422abddeec24dd316b24be76b00b52a18cf287f6af25888244496","family_id":"qf-mod44-exp-addition-convolution","seed":202611132056,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":5,\"x\":[6,7],\"y\":[-7,8]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=6/7, y=-7/8, declared degree=5. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(6/7, -7/8, 5, [[0, -16807/3932160], [1, 343/16384], [2, -21/512], [3, 9/224], [4, -27/1372], [5, 324/84035]], -1/66087813120, -1/66087813120, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (6/7, -7/8, 5, [[0, -16807/3932160], [1, 343/16384], [2, -21/512], [3, 9/224], [4, -27/1372], [5, 324/84035]], -1/66087813120, -1/66087813120, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0057","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71e0c3319e03601e9bd982a2098c1b7a3e3f2d6837746df3a6e900dfcb3e8b13","family_id":"qf-mod44-exp-addition-convolution","seed":202611132057,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":6,\"x\":[7,8],\"y\":[-8,9]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=7/8, y=-8/9, declared degree=6. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0058","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ed7d46782628e80ca461fd0076e01bfac23461b0ca2e382f7f44ad2e27b5472","family_id":"qf-mod44-exp-addition-convolution","seed":202611132058,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":7,\"x\":[8,9],\"y\":[-9,10]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=8/9, y=-9/10, declared degree=7. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(8/9, -9/10, 7, [[0, -531441/5600000000], [1, 6561/10000000], [2, -243/125000], [3, 2/625], [4, -32/10125], [5, 512/273375], [6, -8192/13286025], [7, 131072/1506635235]], -1/241061637600000000, -1/241061637600000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (8/9, -9/10, 7, [[0, -531441/5600000000], [1, 6561/10000000], [2, -243/125000], [3, 2/625], [4, -32/10125], [5, 512/273375], [6, -8192/13286025], [7, 131072/1506635235]], -1/241061637600000000, -1/241061637600000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0059","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45bc35f2b0f694737e2db283633e59e49be33d07c1383ecf04fcc103b4e6d4d1","family_id":"qf-mod44-exp-addition-convolution","seed":202611132059,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":8,\"x\":[9,10],\"y\":[-10,11]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=9/10, y=-10/11, declared degree=8. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(9/10, -10/11, 8, [[0, 156250/13504609503], [1, -12500/136410197], [2, 1125/3543122], [3, -405/644204], [4, 729/937024], [5, -6561/10648000], [6, 59049/193600000], [7, -531441/6160000000], [8, 4782969/448000000000]], 1/864295008192000000000, 1/864295008192000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (9/10, -10/11, 8, [[0, 156250/13504609503], [1, -12500/136410197], [2, 1125/3543122], [3, -405/644204], [4, 729/937024], [5, -6561/10648000], [6, 59049/193600000], [7, -531441/6160000000], [8, 4782969/448000000000]], 1/864295008192000000000, 1/864295008192000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0060","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"779527f3dc0cc31b0989a57ec4e3a2a2c3381568ae298913eb1f31b8a11f0cbb","family_id":"qf-mod44-exp-addition-convolution","seed":202611132060,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":9,\"x\":[10,11],\"y\":[-11,12]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=10/11, y=-11/12, declared degree=9. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(10/11, -11/12, 9, [[0, -2357947691/1872381094133760], [1, 19487171/1733686198272], [2, -805255/18059231232], [3, 33275/322486272], [4, -1375/8957952], [5, 625/4105728], [6, -3125/31049568], [7, 15625/365263668], [8, -78125/7366150638], [9, 1562500/1336956340797]], -1/4414976677584753037148160, -1/4414976677584753037148160, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (10/11, -11/12, 9, [[0, -2357947691/1872381094133760], [1, 19487171/1733686198272], [2, -805255/18059231232], [3, 33275/322486272], [4, -1375/8957952], [5, 625/4105728], [6, -3125/31049568], [7, 15625/365263668], [8, -78125/7366150638], [9, 1562500/1336956340797]], -1/4414976677584753037148160, -1/4414976677584753037148160, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0061","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"272231de47ddbccd82e692b1392f969ede245a947bbffc396b257a4cb13a8594","family_id":"qf-mod44-exp-addition-convolution","seed":202611132061,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":10,\"x\":[11,12],\"y\":[-12,13]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=11/12, y=-12/13, declared degree=10. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(11/12, -12/13, 10, [[0, 2985984/24125236073575], [1, -456192/371157478055], [2, 156816/28550575235], [3, -31944/2196198095], [4, 14641/579217080], [5, -161051/5346619200], [6, 1771561/71068907520], [7, -19487171/1377643438080], [8, 214358881/40693467709440], [9, -2357947691/2028412851978240], [10, 25937424601/224685731296051200]], 1/30974836056463278561086668800, 1/30974836056463278561086668800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (11/12, -12/13, 10, [[0, 2985984/24125236073575], [1, -456192/371157478055], [2, 156816/28550575235], [3, -31944/2196198095], [4, 14641/579217080], [5, -161051/5346619200], [6, 1771561/71068907520], [7, -19487171/1377643438080], [8, 214358881/40693467709440], [9, -2357947691/2028412851978240], [10, 25937424601/224685731296051200]], 1/30974836056463278561086668800, 1/30974836056463278561086668800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0062","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9847229cd31735d5c4027b6ecd76f1361f4f60864f874a4b387292a72313585d","family_id":"qf-mod44-exp-addition-convolution","seed":202611132062,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":11,\"x\":[12,13],\"y\":[-13,14]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=12/13, y=-13/14, declared degree=11. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(12/13, -13/14, 11, [[0, -1792160394037/161645682964443955200], [1, 10604499373/87470607664742400], [2, -62748517/104131675791360], [3, 371293/206610467840], [4, -6591/1844736320], [5, 117/23529800], [6, -27/5462275], [7, 648/184624895], [8, -7776/4457372465], [9, 62208/107613706655], [10, -1492992/12990511731925], [11, 35831808/3449908758521225]], -1/289694990875937857011834775142400, -1/289694990875937857011834775142400, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (12/13, -13/14, 11, [[0, -1792160394037/161645682964443955200], [1, 10604499373/87470607664742400], [2, -62748517/104131675791360], [3, 371293/206610467840], [4, -6591/1844736320], [5, 117/23529800], [6, -27/5462275], [7, 648/184624895], [8, -7776/4457372465], [9, 62208/107613706655], [10, -1492992/12990511731925], [11, 35831808/3449908758521225]], -1/289694990875937857011834775142400, -1/289694990875937857011834775142400, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0063","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30f9c1c5f3d0e3f15d0cce7e8931b2389f6273df1ded0a6b0b518bc8700bd5c0","family_id":"qf-mod44-exp-addition-convolution","seed":202611132063,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":12,\"x\":[13,14],\"y\":[-14,15]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=13/14, y=-14/15, declared degree=12. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(13/14, -14/15, 12, [[0, 7909306972/8670299029541015625], [1, -2098387564/192673311767578125], [2, 139178767/2335434082031250], [3, -36924979/186834726562500], [4, 9796423/22143375000000], [5, -2599051/3690562500000], [6, 4826809/5904900000000], [7, -62748517/90016920000000], [8, 815730721/1881953740800000], [9, -10604499373/55329439979520000], [10, 137858491849/2409904496885760000], [11, -1792160394037/173191803176189952000], [12, 23298085122481/27156474738026584473600]], 1/3523453147278219257625600000000000000, 1/3523453147278219257625600000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (13/14, -14/15, 12, [[0, 7909306972/8670299029541015625], [1, -2098387564/192673311767578125], [2, 139178767/2335434082031250], [3, -36924979/186834726562500], [4, 9796423/22143375000000], [5, -2599051/3690562500000], [6, 4826809/5904900000000], [7, -62748517/90016920000000], [8, 815730721/1881953740800000], [9, -10604499373/55329439979520000], [10, 137858491849/2409904496885760000], [11, -1792160394037/173191803176189952000], [12, 23298085122481/27156474738026584473600]], 1/3523453147278219257625600000000000000, 1/3523453147278219257625600000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0064","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5218813ef06a99b35015a1fc5a33badda68fd5aa0f595f20b6e96fd47523113","family_id":"qf-mod44-exp-addition-convolution","seed":202611132064,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":13,\"x\":[14,15],\"y\":[-15,16]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=14/15, y=-15/16, declared degree=13. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(14/15, -15/16, 13, [[0, -320361328125/4616297704445815291904], [1, 1423828125/1585267068834414592], [2, -132890625/24769797950537728], [3, 1378125/70368744177664], [4, -214375/4398046511104], [5, 12005/137438953472], [6, -16807/144955146240], [7, 117649/1019215872000], [8, -823543/9555148800000], [9, 5764801/120932352000000], [10, -40353607/2125764000000000], [11, 282475249/54804853125000000], [12, -1977326743/2312079741210937500], [13, 110730297608/1690708310760498046875]], -1/54579111145812232494907392000000000000000, -1/54579111145812232494907392000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (14/15, -15/16, 13, [[0, -320361328125/4616297704445815291904], [1, 1423828125/1585267068834414592], [2, -132890625/24769797950537728], [3, 1378125/70368744177664], [4, -214375/4398046511104], [5, 12005/137438953472], [6, -16807/144955146240], [7, 117649/1019215872000], [8, -823543/9555148800000], [9, 5764801/120932352000000], [10, -40353607/2125764000000000], [11, 282475249/54804853125000000], [12, -1977326743/2312079741210937500], [13, 110730297608/1690708310760498046875]], -1/54579111145812232494907392000000000000000, -1/54579111145812232494907392000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0065","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a4489dac0890cdfc298bdb815709f641fdf888442d8fba138227f3978c60782","family_id":"qf-mod44-exp-addition-convolution","seed":202611132065,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":14,\"x\":[15,16],\"y\":[-16,17]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=15/16, y=-16/17, declared degree=14. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(15/16, -16/17, 14, [[0, 35184372088832/7167427341512963030230725], [1, -274877906944/4015365457430231389485], [2, 536870912/1211271631200673119], [3, -41943040/23750424141189669], [4, 204800/42335871909429], [5, -8000/830115135479], [6, 5625/390642416696], [7, -84375/5147288314112], [8, 1265625/88585264431104], [9, -6328125/666994932187136], [10, 94921875/20088317957636096], [11, -1423828125/831892696598577152], [12, 21357421875/50109301253937823744], [13, -320361328125/4904816310973678747648], [14, 4805419921875/1034050685795862625386496]], 1/1057725582686384458266478824154331142212812800, 1/1057725582686384458266478824154331142212812800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (15/16, -16/17, 14, [[0, 35184372088832/7167427341512963030230725], [1, -274877906944/4015365457430231389485], [2, 536870912/1211271631200673119], [3, -41943040/23750424141189669], [4, 204800/42335871909429], [5, -8000/830115135479], [6, 5625/390642416696], [7, -84375/5147288314112], [8, 1265625/88585264431104], [9, -6328125/666994932187136], [10, 94921875/20088317957636096], [11, -1423828125/831892696598577152], [12, 21357421875/50109301253937823744], [13, -320361328125/4904816310973678747648], [14, 4805419921875/1034050685795862625386496]], 1/1057725582686384458266478824154331142212812800, 1/1057725582686384458266478824154331142212812800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0066","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d17669202ab9342d28d43057633bb6def8588c0df7fa97fbd837c09e25496d2","family_id":"qf-mod44-exp-addition-convolution","seed":202611132066,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":15,\"x\":[16,17],\"y\":[-17,18]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=16/17, y=-17/18, declared degree=15. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(16/17, -17/18, 15, [[0, -2862423051509815793/8822409004275290841311870976000], [1, 9904578032905937/2042224306545206213266636800], [2, -34271896307633/1013008088564090383564800], [3, 118587876497/811705199169944217600], [4, -410338673/939473610150398400], [5, 1419857/1482755066525250], [6, -78608/49425168884175], [7, 8704/4271310891225], [8, -16384/8068031683425], [9, 524288/333094450929975], [10, -16777216/17826721540511625], [11, 536870912/1259358839495254575], [12, -17179869184/121318234871376190725], [13, 549755813888/16881207718954086983475], [14, -17592186044416/3794520357271568663063325], [15, 562949953421312/1827693972085805572708834875]], -1/25253466903685353496585230352209316437943123968000, -1/25253466903685353496585230352209316437943123968000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (16/17, -17/18, 15, [[0, -2862423051509815793/8822409004275290841311870976000], [1, 9904578032905937/2042224306545206213266636800], [2, -34271896307633/1013008088564090383564800], [3, 118587876497/811705199169944217600], [4, -410338673/939473610150398400], [5, 1419857/1482755066525250], [6, -78608/49425168884175], [7, 8704/4271310891225], [8, -16384/8068031683425], [9, 524288/333094450929975], [10, -16777216/17826721540511625], [11, 536870912/1259358839495254575], [12, -17179869184/121318234871376190725], [13, 549755813888/16881207718954086983475], [14, -17592186044416/3794520357271568663063325], [15, 562949953421312/1827693972085805572708834875]], -1/25253466903685353496585230352209316437943123968000, -1/25253466903685353496585230352209316437943123968000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0067","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9643656c24d9f850ced3cce6a3ba63da345c648db70e56e3c2c6e05efd88797","family_id":"qf-mod44-exp-addition-convolution","seed":202611132067,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":16,\"x\":[17,18],\"y\":[-18,19]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=17/18, y=-18/19, declared degree=16. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(17/18, -18/19, 16, [[0, 5083731656658/252638623108540190242595875], [1, -4267824106824/13296769637291588960136625], [2, 335893563963/139965996182016725896175], [3, -23498726697/2104751822285965802950], [4, 4931831529/136340199014475517600], [5, -3105227259/35878999740651452000], [6, 217238121/1373358841747424000], [7, -410338673/1821507516422899200], [8, 6975757441/27610219196304998400], [9, -118587876497/529680257739640627200], [10, 2015993900449/12903488985537110016000], [11, -34271896307633/403403813547844386816000], [12, 582622237229761/16509831863937041852006400], [13, -9904578032905937/914992260669773951061196800], [14, 168377826559400929/72814120954352537579185766400], [15, -2862423051509815793/9312542837846140332495863808000], [16, 48661191875666868481/2540853793231283762297818841088000]], 1/732887459788283721221778416595587951601772740476928000, 1/732887459788283721221778416595587951601772740476928000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (17/18, -18/19, 16, [[0, 5083731656658/252638623108540190242595875], [1, -4267824106824/13296769637291588960136625], [2, 335893563963/139965996182016725896175], [3, -23498726697/2104751822285965802950], [4, 4931831529/136340199014475517600], [5, -3105227259/35878999740651452000], [6, 217238121/1373358841747424000], [7, -410338673/1821507516422899200], [8, 6975757441/27610219196304998400], [9, -118587876497/529680257739640627200], [10, 2015993900449/12903488985537110016000], [11, -34271896307633/403403813547844386816000], [12, 582622237229761/16509831863937041852006400], [13, -9904578032905937/914992260669773951061196800], [14, 168377826559400929/72814120954352537579185766400], [15, -2862423051509815793/9312542837846140332495863808000], [16, 48661191875666868481/2540853793231283762297818841088000]], 1/732887459788283721221778416595587951601772740476928000, 1/732887459788283721221778416595587951601772740476928000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0068","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9745cbce8209ac71a2cb1f8e533760416b825c90f6469a48630080b74259165e","family_id":"qf-mod44-exp-addition-convolution","seed":202611132068,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":17,\"x\":[18,19],\"y\":[-19,20]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=18/19, y=-19/20, declared degree=17. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(18/19, -19/20, 17, [[0, -5480386857784802185939/4662066257539891200000000000000000000], [1, 15181127029874798299/761775532277760000000000000000000], [2, -42052983462257059/264505393152000000000000000000], [3, 116490258898219/146947440640000000000000000], [4, -2904189280011/1049624576000000000000000], [5, 72403610859/10092544000000000000000], [6, -1805076171/126156800000000000000], [7, 45001899/2007040000000000000], [8, -1121931/40140800000000000], [9, 531441/19066880000000000], [10, -4782969/215098240000000000], [11, 43046721/3050553968000000000], [12, -387420489/55062499122400000000], [13, 3486784401/1292041541907116000000], [14, -31381059609/40812362204991026650000], [15, 282429536481/1841657844500220077581250], [16, -2541865828329/132967696372915889601366250], [17, 91507169819844/81602275264058481448358467625]], -1/25549926647943596675599717954252229836800000000000000000000, -1/25549926647943596675599717954252229836800000000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (18/19, -19/20, 17, [[0, -5480386857784802185939/4662066257539891200000000000000000000], [1, 15181127029874798299/761775532277760000000000000000000], [2, -42052983462257059/264505393152000000000000000000], [3, 116490258898219/146947440640000000000000000], [4, -2904189280011/1049624576000000000000000], [5, 72403610859/10092544000000000000000], [6, -1805076171/126156800000000000000], [7, 45001899/2007040000000000000], [8, -1121931/40140800000000000], [9, 531441/19066880000000000], [10, -4782969/215098240000000000], [11, 43046721/3050553968000000000], [12, -387420489/55062499122400000000], [13, 3486784401/1292041541907116000000], [14, -31381059609/40812362204991026650000], [15, 282429536481/1841657844500220077581250], [16, -2541865828329/132967696372915889601366250], [17, 91507169819844/81602275264058481448358467625]], -1/25549926647943596675599717954252229836800000000000000000000, -1/25549926647943596675599717954252229836800000000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0069","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6a3353662dbbb75f71a71548cd65c303de2eaf033a4bbe196d50f63e67731fa","family_id":"qf-mod44-exp-addition-convolution","seed":202611132069,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":18,\"x\":[19,20],\"y\":[-20,21]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=19/20, y=-20/21, declared degree=18. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(19/20, -20/21, 18, [[0, 32000000000000000/493058422447458087543290824655799], [1, -3040000000000000/2608774721944222685414237167491], [2, 72200000000000/7307492218331155981552485063], [3, -54872000000000/1043927459761593711650355009], [4, 3258025000000/16570277139072916057942143], [5, -61902475000/112722973735189905156069], [6, 5880735125/4954855988359995831036], [7, -4469358695/2202158217048887036016], [8, 16983563041/6101217571044535511040], [9, -322687697779/104592301217906323046400], [10, 6131066257801/2213593676569445990400000], [11, -116490258898219/57975072481580728320000000], [12, 2213314919066161/1893063591235289088000000000], [13, -42052983462257059/78126433923996057600000000000], [14, 799006685782884121/4166743142613123072000000000000], [15, -15181127029874798299/297624510186651648000000000000000], [16, 288441413567621167681/30234870876104294400000000000000000], [17, -5480386857784802185939/4895169570416885760000000000000000000], [18, 104127350297911241532841/1678343852714360832000000000000000000000]], 1/1058834899714592382164569538056763580874752000000000000000000000, 1/1058834899714592382164569538056763580874752000000000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (19/20, -20/21, 18, [[0, 32000000000000000/493058422447458087543290824655799], [1, -3040000000000000/2608774721944222685414237167491], [2, 72200000000000/7307492218331155981552485063], [3, -54872000000000/1043927459761593711650355009], [4, 3258025000000/16570277139072916057942143], [5, -61902475000/112722973735189905156069], [6, 5880735125/4954855988359995831036], [7, -4469358695/2202158217048887036016], [8, 16983563041/6101217571044535511040], [9, -322687697779/104592301217906323046400], [10, 6131066257801/2213593676569445990400000], [11, -116490258898219/57975072481580728320000000], [12, 2213314919066161/1893063591235289088000000000], [13, -42052983462257059/78126433923996057600000000000], [14, 799006685782884121/4166743142613123072000000000000], [15, -15181127029874798299/297624510186651648000000000000000], [16, 288441413567621167681/30234870876104294400000000000000000], [17, -5480386857784802185939/4895169570416885760000000000000000000], [18, 104127350297911241532841/1678343852714360832000000000000000000000]], 1/1058834899714592382164569538056763580874752000000000000000000000, 1/1058834899714592382164569538056763580874752000000000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0070","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e81885ff0e551b33ec5d7140c25690f4fc4359feeca5722cc3dc6fe1d7ee9529","family_id":"qf-mod44-exp-addition-convolution","seed":202611132070,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":19,\"x\":[20,21],\"y\":[-21,22]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=20/21, y=-21/22, declared degree=19. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(20/21, -21/22, 19, [[0, -41209797661291758429/12132755312575326542353091425992704000], [1, 93446253200208069/1451286520642981643822140122726400], [2, -1907066391840981/3298378456006776463232136642560], [3, 1441471195647/440959686631921987063119872], [4, -16343210835/1252726382477051099611136], [5, 555891525/14235527073602853404672], [6, -14706125/161767353109123334144], [7, 214375/1272645260473872384], [8, -21875/86771267759582208], [9, 15625/50825732251738752], [10, -78125/254706226397918064], [11, 390625/1560075636687248142], [12, -15625000/93817275788055876903], [13, 625000000/6985122624583433016687], [14, -25000000000/653426470972395688560993], [15, 1000000000000/78589383736043590542381249], [16, -10000000000000/3150719838872293039017284619], [17, 400000000000000/715786261576532754954926751171], [18, -16000000000000000/258268697472478045856009479581609], [19, 640000000000000000/196730310556535776929773039037663801]], -1/51676388762462995371436759840422682365878499503552553575761575936000, -1/51676388762462995371436759840422682365878499503552553575761575936000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (20/21, -21/22, 19, [[0, -41209797661291758429/12132755312575326542353091425992704000], [1, 93446253200208069/1451286520642981643822140122726400], [2, -1907066391840981/3298378456006776463232136642560], [3, 1441471195647/440959686631921987063119872], [4, -16343210835/1252726382477051099611136], [5, 555891525/14235527073602853404672], [6, -14706125/161767353109123334144], [7, 214375/1272645260473872384], [8, -21875/86771267759582208], [9, 15625/50825732251738752], [10, -78125/254706226397918064], [11, 390625/1560075636687248142], [12, -15625000/93817275788055876903], [13, 625000000/6985122624583433016687], [14, -25000000000/653426470972395688560993], [15, 1000000000000/78589383736043590542381249], [16, -10000000000000/3150719838872293039017284619], [17, 400000000000000/715786261576532754954926751171], [18, -16000000000000000/258268697472478045856009479581609], [19, 640000000000000000/196730310556535776929773039037663801]], -1/51676388762462995371436759840422682365878499503552553575761575936000, -1/51676388762462995371436759840422682365878499503552553575761575936000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0071","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a56956078ccdac9f5c72a87015c38b85f59e520a695cc1b61a9237d776d7e748","family_id":"qf-mod44-exp-addition-convolution","seed":202611132071,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":20,\"x\":[21,22],\"y\":[-22,23]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=21/22, y=-22/23, declared degree=20. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(21/22, -22/23, 20, [[0, 244636361793658185164/1447933879643515222582682716129837194375], [1, -2021788113997175084/599558542295451437922435907300139625], [2, 4177248169415651/130665477235578388999114287305250], [3, -241658985007517/1262468379087713903373084901500], [4, 13980271859939/17220370047232244206282488000], [5, -808776058013/311963225493337757360190000], [6, 46788697571/7233929866512179880816000], [7, -2706784157/209679126565570431328000], [8, 469772457/22440575418388808704000], [9, -27176919/975677192103861248000], [10, 51883209/1696829899311063040000], [11, -1089547389/39277923408400433152000], [12, 68641485507/3306176509507097329664000], [13, -1441471195647/113056861944666610860032000], [14, 30270895108587/4758219233149446926630912000], [15, -635688797280327/250323707483079599183626240000], [16, 13349464742886867/16856580789121464487634796544000], [17, -280338759600624207/1507564638400993584828903325696000], [18, 1962371317204369449/63448807390093990874538192142336000], [19, -41209797661291758429/12684244190419659567005504672628736000], [20, 865405750887126927009/5338412337533143678635360227436789760000]], 1/2945336732372916073844860985203962504337568772809813725658735066480640000, 1/2945336732372916073844860985203962504337568772809813725658735066480640000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (21/22, -22/23, 20, [[0, 244636361793658185164/1447933879643515222582682716129837194375], [1, -2021788113997175084/599558542295451437922435907300139625], [2, 4177248169415651/130665477235578388999114287305250], [3, -241658985007517/1262468379087713903373084901500], [4, 13980271859939/17220370047232244206282488000], [5, -808776058013/311963225493337757360190000], [6, 46788697571/7233929866512179880816000], [7, -2706784157/209679126565570431328000], [8, 469772457/22440575418388808704000], [9, -27176919/975677192103861248000], [10, 51883209/1696829899311063040000], [11, -1089547389/39277923408400433152000], [12, 68641485507/3306176509507097329664000], [13, -1441471195647/113056861944666610860032000], [14, 30270895108587/4758219233149446926630912000], [15, -635688797280327/250323707483079599183626240000], [16, 13349464742886867/16856580789121464487634796544000], [17, -280338759600624207/1507564638400993584828903325696000], [18, 1962371317204369449/63448807390093990874538192142336000], [19, -41209797661291758429/12684244190419659567005504672628736000], [20, 865405750887126927009/5338412337533143678635360227436789760000]], 1/2945336732372916073844860985203962504337568772809813725658735066480640000, 1/2945336732372916073844860985203962504337568772809813725658735066480640000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0072","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dd75bdecf1bc6499afcfc2648cda47227a393480c77ed7a410edc529cb23b40","family_id":"qf-mod44-exp-addition-convolution","seed":202611132072,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":0,\"x\":[22,23],\"y\":[-23,24]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=22/23, y=-23/24, declared degree=0. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(22/23, -23/24, 0, [[0, 1]], 1, 1, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (22/23, -23/24, 0, [[0, 1]], 1, 1, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0073","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f909099bdf03a8d61c5f72e1ffe91474f7b62a01377f669b45454d742d3c91df","family_id":"qf-mod44-exp-addition-convolution","seed":202611132073,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":1,\"x\":[23,24],\"y\":[-24,25]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=23/24, y=-24/25, declared degree=1. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(23/24, -24/25, 1, [[0, -24/25], [1, 23/24]], -1/600, -1/600, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (23/24, -24/25, 1, [[0, -24/25], [1, 23/24]], -1/600, -1/600, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0074","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25a198785806c955a32070bad1e14c9515d160362f1cb99cbabb50a26f18cbd5","family_id":"qf-mod44-exp-addition-convolution","seed":202611132074,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":2,\"x\":[24,25],\"y\":[-25,26]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=24/25, y=-25/26, declared degree=2. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(24/25, -25/26, 2, [[0, 625/1352], [1, -12/13], [2, 288/625]], 1/845000, 1/845000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (24/25, -25/26, 2, [[0, 625/1352], [1, -12/13], [2, 288/625]], 1/845000, 1/845000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0075","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb50d60cc8a5d3fbf057b5a8aad201c4838814edbb8de40c4c1fe6d0f57a8b1c","family_id":"qf-mod44-exp-addition-convolution","seed":202611132075,"semantic_template":"qf-mod44-exp-addition-convolution/{\"degree\":3,\"x\":[25,26],\"y\":[-26,27]}"},"fields":{"title":"One Convolution Degree Is Not the Whole Exponential Addition Proof","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=25/26, y=-26/27, declared degree=3. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).","answer":{"kind":"expression","value":"(25/26, -26/27, 3, [[0, -8788/59049], [1, 325/729], [2, -625/1404], [3, 15625/105456]], -1/2075690448, -1/2075690448, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)","explanation":"Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (25/26, -26/27, 3, [[0, -8788/59049], [1, 325/729], [2, -625/1404], [3, 15625/105456]], -1/2075690448, -1/2075690448, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)."},"skills":["One Convolution Degree Is Not the Whole Exponential Addition Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0076","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"612f1f4b8cf5ba80525fccb68156e02690c7aad94d38d955a63e04bd04fa8494","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132076,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-12,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-12. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-12, E(0)=1, E(-12)*E(12)=1, E(-6)*E(6)=1, E(-6)!=0, E(-12)=E(-6)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-12, E(0)=1, E(-12)*E(12)=1, E(-6)*E(6)=1, E(-6)!=0, E(-12)=E(-6)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0077","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a443cf97f58e484e8995db42312ff41e64e7af58d686ead7251ea6350ce2ea7","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132077,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-11,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-11. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-11, E(0)=1, E(-11)*E(11)=1, E(-11/2)*E(11/2)=1, E(-11/2)!=0, E(-11)=E(-11/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-11, E(0)=1, E(-11)*E(11)=1, E(-11/2)*E(11/2)=1, E(-11/2)!=0, E(-11)=E(-11/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0078","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ec5932f684e0a3d57ca6f0440aa48ad99ae451d753b1a0b05f76808a9d0f31c","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132078,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-10,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-10. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-10, E(0)=1, E(-10)*E(10)=1, E(-5)*E(5)=1, E(-5)!=0, E(-10)=E(-5)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-10, E(0)=1, E(-10)*E(10)=1, E(-5)*E(5)=1, E(-5)!=0, E(-10)=E(-5)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0079","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df61b466645e88862734c39f5598f30a46e2dbca2c10e274aa4d2129238d563b","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132079,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-9,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-9. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-9, E(0)=1, E(-9)*E(9)=1, E(-9/2)*E(9/2)=1, E(-9/2)!=0, E(-9)=E(-9/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-9, E(0)=1, E(-9)*E(9)=1, E(-9/2)*E(9/2)=1, E(-9/2)!=0, E(-9)=E(-9/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0080","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6e0afaec1daeb18cc5310e87b265848ba141f33debdd215be243d8ec812e73b","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132080,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-8,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-8. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-8, E(0)=1, E(-8)*E(8)=1, E(-4)*E(4)=1, E(-4)!=0, E(-8)=E(-4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-8, E(0)=1, E(-8)*E(8)=1, E(-4)*E(4)=1, E(-4)!=0, E(-8)=E(-4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0081","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b50320404798ed04986da9ebcdc3f982b1fe0048a10ccf5e89e75ab5f454394","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132081,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-7,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-7. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-7, E(0)=1, E(-7)*E(7)=1, E(-7/2)*E(7/2)=1, E(-7/2)!=0, E(-7)=E(-7/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-7, E(0)=1, E(-7)*E(7)=1, E(-7/2)*E(7/2)=1, E(-7/2)!=0, E(-7)=E(-7/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0082","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f54a5d2a74e8517f1c3db84cd27903b1ec4412b983419f953c6ac963cfcc84","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132082,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-6,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-6. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-6, E(0)=1, E(-6)*E(6)=1, E(-3)*E(3)=1, E(-3)!=0, E(-6)=E(-3)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-6, E(0)=1, E(-6)*E(6)=1, E(-3)*E(3)=1, E(-3)!=0, E(-6)=E(-3)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0083","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf0bdf5e04cb3e656cd59be2f84a2d6810aecc13c0ecdd9dcb855d0416a63805","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132083,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-5,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-5. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-5, E(0)=1, E(-5)*E(5)=1, E(-5/2)*E(5/2)=1, E(-5/2)!=0, E(-5)=E(-5/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-5, E(0)=1, E(-5)*E(5)=1, E(-5/2)*E(5/2)=1, E(-5/2)!=0, E(-5)=E(-5/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0084","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f88da3f566d1bc1628ad9261b5e84e4250e249535e6047494418baaeddcb0df","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132084,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-4,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-4. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-4, E(0)=1, E(-4)*E(4)=1, E(-2)*E(2)=1, E(-2)!=0, E(-4)=E(-2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-4, E(0)=1, E(-4)*E(4)=1, E(-2)*E(2)=1, E(-2)!=0, E(-4)=E(-2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0085","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b23546808547a0bd7462a6bb51f3416bf7b1faef9439b571167768ee7f74a899","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132085,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-3,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-3. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-3, E(0)=1, E(-3)*E(3)=1, E(-3/2)*E(3/2)=1, E(-3/2)!=0, E(-3)=E(-3/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-3, E(0)=1, E(-3)*E(3)=1, E(-3/2)*E(3/2)=1, E(-3/2)!=0, E(-3)=E(-3/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0086","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c311033b6e5a83d16913effd70843c0f9839fa6494dae6a7b5c7fec7f077e7af","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132086,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-2,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-2. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-2, E(0)=1, E(-2)*E(2)=1, E(-1)*E(1)=1, E(-1)!=0, E(-2)=E(-1)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-2, E(0)=1, E(-2)*E(2)=1, E(-1)*E(1)=1, E(-1)!=0, E(-2)=E(-1)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0087","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d069931e63971abadadea7de9be8f9bfc05fb920e4be01e0d7542594c0830862","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132087,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[-1,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-1. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(-1, E(0)=1, E(-1)*E(1)=1, E(-1/2)*E(1/2)=1, E(-1/2)!=0, E(-1)=E(-1/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-1, E(0)=1, E(-1)*E(1)=1, E(-1/2)*E(1/2)=1, E(-1/2)!=0, E(-1)=E(-1/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0088","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7862feeb90224d40d74fd3d745afef759bc1fa464823f96089639beb5dc20fe9","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132088,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[0,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=0. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(0, E(0)=1, E(0)*E(0)=1, E(0)*E(0)=1, E(0)!=0, E(0)=E(0)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (0, E(0)=1, E(0)*E(0)=1, E(0)*E(0)=1, E(0)!=0, E(0)=E(0)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0089","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52c509a155385508cb5508bd241076004f93130fd565ca74a1811d873c2458e0","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132089,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[1,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=1. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(1, E(0)=1, E(1)*E(-1)=1, E(1/2)*E(-1/2)=1, E(1/2)!=0, E(1)=E(1/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (1, E(0)=1, E(1)*E(-1)=1, E(1/2)*E(-1/2)=1, E(1/2)!=0, E(1)=E(1/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0090","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22ad55146d0c896be18370384577ff15052ea04da199aef5372ef33fde48a3d5","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132090,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[2,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=2. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(2, E(0)=1, E(2)*E(-2)=1, E(1)*E(-1)=1, E(1)!=0, E(2)=E(1)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (2, E(0)=1, E(2)*E(-2)=1, E(1)*E(-1)=1, E(1)!=0, E(2)=E(1)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0091","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f35c3c5e51b06411f1d31c52d33e11d29a484f0ba2bfd658505aeab0bbca1c24","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132091,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[3,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=3. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(3, E(0)=1, E(3)*E(-3)=1, E(3/2)*E(-3/2)=1, E(3/2)!=0, E(3)=E(3/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (3, E(0)=1, E(3)*E(-3)=1, E(3/2)*E(-3/2)=1, E(3/2)!=0, E(3)=E(3/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0092","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"702b3a0406de73277f6e12c6554f7fa51a55f6258a825d8d7fee90d9a6fbec43","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132092,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[4,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=4. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(4, E(0)=1, E(4)*E(-4)=1, E(2)*E(-2)=1, E(2)!=0, E(4)=E(2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (4, E(0)=1, E(4)*E(-4)=1, E(2)*E(-2)=1, E(2)!=0, E(4)=E(2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0093","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d8af4fce1427bc67b9619e849fed05d95cfd0c70e99adcb261bec33cb071805","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132093,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[5,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=5. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(5, E(0)=1, E(5)*E(-5)=1, E(5/2)*E(-5/2)=1, E(5/2)!=0, E(5)=E(5/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (5, E(0)=1, E(5)*E(-5)=1, E(5/2)*E(-5/2)=1, E(5/2)!=0, E(5)=E(5/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0094","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96990afcd5e1e953bcd06373a68feb028afda39a4a574e7d86ce2b0adaf37c34","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132094,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[6,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=6. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(6, E(0)=1, E(6)*E(-6)=1, E(3)*E(-3)=1, E(3)!=0, E(6)=E(3)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (6, E(0)=1, E(6)*E(-6)=1, E(3)*E(-3)=1, E(3)!=0, E(6)=E(3)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0095","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ea5738dbbb31123a34bd847e44d1f103555382c0d04b88f2fc002523ab61f2f","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132095,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[7,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=7. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(7, E(0)=1, E(7)*E(-7)=1, E(7/2)*E(-7/2)=1, E(7/2)!=0, E(7)=E(7/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (7, E(0)=1, E(7)*E(-7)=1, E(7/2)*E(-7/2)=1, E(7/2)!=0, E(7)=E(7/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0096","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9becc5883174f1417e905db171fb872aa673f46c444a7e0aff12a95c7017aa96","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132096,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[8,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=8. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(8, E(0)=1, E(8)*E(-8)=1, E(4)*E(-4)=1, E(4)!=0, E(8)=E(4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (8, E(0)=1, E(8)*E(-8)=1, E(4)*E(-4)=1, E(4)!=0, E(8)=E(4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0097","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16776681a4c90af54644dab89e529f40b31ec3d8513fc4d46d5cef07f926dd15","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132097,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[9,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=9. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(9, E(0)=1, E(9)*E(-9)=1, E(9/2)*E(-9/2)=1, E(9/2)!=0, E(9)=E(9/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (9, E(0)=1, E(9)*E(-9)=1, E(9/2)*E(-9/2)=1, E(9/2)!=0, E(9)=E(9/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0098","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e5b34f6ed4b984d990208998c3583357d3f0ad295d86768acf4f8e8e28d6051","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132098,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[10,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=10. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(10, E(0)=1, E(10)*E(-10)=1, E(5)*E(-5)=1, E(5)!=0, E(10)=E(5)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (10, E(0)=1, E(10)*E(-10)=1, E(5)*E(-5)=1, E(5)!=0, E(10)=E(5)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0099","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aa9aa36eace37c6dfa27c37fb4e50ff6ec2ad2585d3ade69b28995a8ddb5cd5","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132099,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[11,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=11. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(11, E(0)=1, E(11)*E(-11)=1, E(11/2)*E(-11/2)=1, E(11/2)!=0, E(11)=E(11/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (11, E(0)=1, E(11)*E(-11)=1, E(11/2)*E(-11/2)=1, E(11/2)!=0, E(11)=E(11/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r02-real-exponential-function-0100","batch_id":"batch-20261015-advance-20261113-r02-real-exponential-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d5526559e0f7499e2e1b1a8a48a0b3fbeecb7bb4ac1beddcc954cd0882a1b90","family_id":"qf-mod44-exp-zero-inverse-positivity","seed":202611132100,"semantic_template":"qf-mod44-exp-zero-inverse-positivity/{\"x\":[12,1]}"},"fields":{"title":"Inverse and Half-Value Nonvanishing Precede Strict Square Positivity","stem":"Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=12. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).","answer":{"kind":"expression","value":"(12, E(0)=1, E(12)*E(-12)=1, E(6)*E(-6)=1, E(6)!=0, E(12)=E(6)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)","explanation":"Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (12, E(0)=1, E(12)*E(-12)=1, E(6)*E(-6)=1, E(6)!=0, E(12)=E(6)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)."},"skills":["Inverse and Half-Value Nonvanishing Precede Strict Square Positivity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}