第 24211 題：實指數函數

原題與已知條件

E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[11,12],"y":[-12,13],"degree":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[11,12],"y":[-12,13],"degree":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-real-exponential-function

逐步推導

source roles = [exp-addition-convolution]

derived = [[x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate]]

(11/12)^0

Exp convolution x power0 = 1

(-12/13)^10

Exp convolution y power10 = 61917364224/137858491849

Exp convolution numerator0 := mul(1, 61917364224/137858491849)

Exp convolution numerator0 = 61917364224/137858491849

source roles = [0]

derived = [1]

Exp convolution right0 factorial step1 := mul(1, 1)

Exp convolution right0 factorial step1 = 1

Exp convolution right0 factorial step2 := mul(1, 2)

Exp convolution right0 factorial step2 = 2

Exp convolution right0 factorial step3 := mul(2, 3)

Exp convolution right0 factorial step3 = 6

Exp convolution right0 factorial step4 := mul(6, 4)

Exp convolution right0 factorial step4 = 24

Exp convolution right0 factorial step5 := mul(24, 5)

Exp convolution right0 factorial step5 = 120

Exp convolution right0 factorial step6 := mul(120, 6)

Exp convolution right0 factorial step6 = 720

Exp convolution right0 factorial step7 := mul(720, 7)

Exp convolution right0 factorial step7 = 5040

Exp convolution right0 factorial step8 := mul(5040, 8)

Exp convolution right0 factorial step8 = 40320

Exp convolution right0 factorial step9 := mul(40320, 9)

Exp convolution right0 factorial step9 = 362880

Exp convolution right0 factorial step10 := mul(362880, 10)

Exp convolution right0 factorial step10 = 3628800

Exp convolution denominator0 := mul(1, 3628800)

Exp convolution denominator0 = 3628800

Exp convolution coefficient term0 := div(61917364224/137858491849, 3628800)

Exp convolution coefficient term0 = 2985984/24125236073575

(11/12)^1

Exp convolution x power1 = 11/12

(-12/13)^9

Exp convolution y power9 = -5159780352/10604499373

Exp convolution numerator1 := mul(11/12, -5159780352/10604499373)

Exp convolution numerator1 = -4729798656/10604499373

Exp convolution left1 factorial step1 := mul(1, 1)

Exp convolution left1 factorial step1 = 1

Exp convolution right1 factorial step1 := mul(1, 1)

Exp convolution right1 factorial step1 = 1

Exp convolution right1 factorial step2 := mul(1, 2)

Exp convolution right1 factorial step2 = 2

Exp convolution right1 factorial step3 := mul(2, 3)

Exp convolution right1 factorial step3 = 6

Exp convolution right1 factorial step4 := mul(6, 4)

Exp convolution right1 factorial step4 = 24

Exp convolution right1 factorial step5 := mul(24, 5)

Exp convolution right1 factorial step5 = 120

Exp convolution right1 factorial step6 := mul(120, 6)

Exp convolution right1 factorial step6 = 720

Exp convolution right1 factorial step7 := mul(720, 7)

Exp convolution right1 factorial step7 = 5040

Exp convolution right1 factorial step8 := mul(5040, 8)

Exp convolution right1 factorial step8 = 40320

Exp convolution right1 factorial step9 := mul(40320, 9)

Exp convolution right1 factorial step9 = 362880

Exp convolution denominator1 := mul(1, 362880)

Exp convolution denominator1 = 362880

Exp convolution coefficient term1 := div(-4729798656/10604499373, 362880)

Exp convolution coefficient term1 = -456192/371157478055

(11/12)^2

Exp convolution x power2 = 121/144

(-12/13)^8

Exp convolution y power8 = 429981696/815730721

Exp convolution numerator2 := mul(121/144, 429981696/815730721)

Exp convolution numerator2 = 361304064/815730721

Exp convolution left2 factorial step1 := mul(1, 1)

Exp convolution left2 factorial step1 = 1

Exp convolution left2 factorial step2 := mul(1, 2)

Exp convolution left2 factorial step2 = 2

Exp convolution right2 factorial step1 := mul(1, 1)

Exp convolution right2 factorial step1 = 1

Exp convolution right2 factorial step2 := mul(1, 2)

Exp convolution right2 factorial step2 = 2

Exp convolution right2 factorial step3 := mul(2, 3)

Exp convolution right2 factorial step3 = 6

Exp convolution right2 factorial step4 := mul(6, 4)

Exp convolution right2 factorial step4 = 24

Exp convolution right2 factorial step5 := mul(24, 5)

Exp convolution right2 factorial step5 = 120

Exp convolution right2 factorial step6 := mul(120, 6)

Exp convolution right2 factorial step6 = 720

Exp convolution right2 factorial step7 := mul(720, 7)

Exp convolution right2 factorial step7 = 5040

Exp convolution right2 factorial step8 := mul(5040, 8)

Exp convolution right2 factorial step8 = 40320

Exp convolution denominator2 := mul(2, 40320)

Exp convolution denominator2 = 80640

Exp convolution coefficient term2 := div(361304064/815730721, 80640)

Exp convolution coefficient term2 = 156816/28550575235

(11/12)^3

Exp convolution x power3 = 1331/1728

(-12/13)^7

Exp convolution y power7 = -35831808/62748517

Exp convolution numerator3 := mul(1331/1728, -35831808/62748517)

Exp convolution numerator3 = -27599616/62748517

Exp convolution left3 factorial step1 := mul(1, 1)

Exp convolution left3 factorial step1 = 1

Exp convolution left3 factorial step2 := mul(1, 2)

Exp convolution left3 factorial step2 = 2

Exp convolution left3 factorial step3 := mul(2, 3)

Exp convolution left3 factorial step3 = 6

Exp convolution right3 factorial step1 := mul(1, 1)

Exp convolution right3 factorial step1 = 1

Exp convolution right3 factorial step2 := mul(1, 2)

Exp convolution right3 factorial step2 = 2

Exp convolution right3 factorial step3 := mul(2, 3)

Exp convolution right3 factorial step3 = 6

Exp convolution right3 factorial step4 := mul(6, 4)

Exp convolution right3 factorial step4 = 24

Exp convolution right3 factorial step5 := mul(24, 5)

Exp convolution right3 factorial step5 = 120

Exp convolution right3 factorial step6 := mul(120, 6)

Exp convolution right3 factorial step6 = 720

Exp convolution right3 factorial step7 := mul(720, 7)

Exp convolution right3 factorial step7 = 5040

Exp convolution denominator3 := mul(6, 5040)

Exp convolution denominator3 = 30240

Exp convolution coefficient term3 := div(-27599616/62748517, 30240)

Exp convolution coefficient term3 = -31944/2196198095

(11/12)^4

Exp convolution x power4 = 14641/20736

(-12/13)^6

Exp convolution y power6 = 2985984/4826809

Exp convolution numerator4 := mul(14641/20736, 2985984/4826809)

Exp convolution numerator4 = 2108304/4826809

Exp convolution left4 factorial step1 := mul(1, 1)

Exp convolution left4 factorial step1 = 1

Exp convolution left4 factorial step2 := mul(1, 2)

Exp convolution left4 factorial step2 = 2

Exp convolution left4 factorial step3 := mul(2, 3)

Exp convolution left4 factorial step3 = 6

Exp convolution left4 factorial step4 := mul(6, 4)

Exp convolution left4 factorial step4 = 24

Exp convolution right4 factorial step1 := mul(1, 1)

Exp convolution right4 factorial step1 = 1

Exp convolution right4 factorial step2 := mul(1, 2)

Exp convolution right4 factorial step2 = 2

Exp convolution right4 factorial step3 := mul(2, 3)

Exp convolution right4 factorial step3 = 6

Exp convolution right4 factorial step4 := mul(6, 4)

Exp convolution right4 factorial step4 = 24

Exp convolution right4 factorial step5 := mul(24, 5)

Exp convolution right4 factorial step5 = 120

Exp convolution right4 factorial step6 := mul(120, 6)

Exp convolution right4 factorial step6 = 720

Exp convolution denominator4 := mul(24, 720)

Exp convolution denominator4 = 17280

Exp convolution coefficient term4 := div(2108304/4826809, 17280)

Exp convolution coefficient term4 = 14641/579217080

(11/12)^5

Exp convolution x power5 = 161051/248832

(-12/13)^5

Exp convolution y power5 = -248832/371293

Exp convolution numerator5 := mul(161051/248832, -248832/371293)

Exp convolution numerator5 = -161051/371293

Exp convolution left5 factorial step1 := mul(1, 1)

Exp convolution left5 factorial step1 = 1

Exp convolution left5 factorial step2 := mul(1, 2)

Exp convolution left5 factorial step2 = 2

Exp convolution left5 factorial step3 := mul(2, 3)

Exp convolution left5 factorial step3 = 6

Exp convolution left5 factorial step4 := mul(6, 4)

Exp convolution left5 factorial step4 = 24

Exp convolution left5 factorial step5 := mul(24, 5)

Exp convolution left5 factorial step5 = 120

Exp convolution right5 factorial step1 := mul(1, 1)

Exp convolution right5 factorial step1 = 1

Exp convolution right5 factorial step2 := mul(1, 2)

Exp convolution right5 factorial step2 = 2

Exp convolution right5 factorial step3 := mul(2, 3)

Exp convolution right5 factorial step3 = 6

Exp convolution right5 factorial step4 := mul(6, 4)

Exp convolution right5 factorial step4 = 24

Exp convolution right5 factorial step5 := mul(24, 5)

Exp convolution right5 factorial step5 = 120

Exp convolution denominator5 := mul(120, 120)

Exp convolution denominator5 = 14400

Exp convolution coefficient term5 := div(-161051/371293, 14400)

Exp convolution coefficient term5 = -161051/5346619200

(11/12)^6

Exp convolution x power6 = 1771561/2985984

(-12/13)^4

Exp convolution y power4 = 20736/28561

Exp convolution numerator6 := mul(1771561/2985984, 20736/28561)

Exp convolution numerator6 = 1771561/4112784

Exp convolution left6 factorial step1 := mul(1, 1)

Exp convolution left6 factorial step1 = 1

Exp convolution left6 factorial step2 := mul(1, 2)

Exp convolution left6 factorial step2 = 2

Exp convolution left6 factorial step3 := mul(2, 3)

Exp convolution left6 factorial step3 = 6

Exp convolution left6 factorial step4 := mul(6, 4)

Exp convolution left6 factorial step4 = 24

Exp convolution left6 factorial step5 := mul(24, 5)

Exp convolution left6 factorial step5 = 120

Exp convolution left6 factorial step6 := mul(120, 6)

Exp convolution left6 factorial step6 = 720

Exp convolution right6 factorial step1 := mul(1, 1)

Exp convolution right6 factorial step1 = 1

Exp convolution right6 factorial step2 := mul(1, 2)

Exp convolution right6 factorial step2 = 2

Exp convolution right6 factorial step3 := mul(2, 3)

Exp convolution right6 factorial step3 = 6

Exp convolution right6 factorial step4 := mul(6, 4)

Exp convolution right6 factorial step4 = 24

Exp convolution denominator6 := mul(720, 24)

Exp convolution denominator6 = 17280

Exp convolution coefficient term6 := div(1771561/4112784, 17280)

Exp convolution coefficient term6 = 1771561/71068907520

(11/12)^7

Exp convolution x power7 = 19487171/35831808

(-12/13)^3

Exp convolution y power3 = -1728/2197

Exp convolution numerator7 := mul(19487171/35831808, -1728/2197)

Exp convolution numerator7 = -19487171/45556992

Exp convolution left7 factorial step1 := mul(1, 1)

Exp convolution left7 factorial step1 = 1

Exp convolution left7 factorial step2 := mul(1, 2)

Exp convolution left7 factorial step2 = 2

Exp convolution left7 factorial step3 := mul(2, 3)

Exp convolution left7 factorial step3 = 6

Exp convolution left7 factorial step4 := mul(6, 4)

Exp convolution left7 factorial step4 = 24

Exp convolution left7 factorial step5 := mul(24, 5)

Exp convolution left7 factorial step5 = 120

Exp convolution left7 factorial step6 := mul(120, 6)

Exp convolution left7 factorial step6 = 720

Exp convolution left7 factorial step7 := mul(720, 7)

Exp convolution left7 factorial step7 = 5040

Exp convolution right7 factorial step1 := mul(1, 1)

Exp convolution right7 factorial step1 = 1

Exp convolution right7 factorial step2 := mul(1, 2)

Exp convolution right7 factorial step2 = 2

Exp convolution right7 factorial step3 := mul(2, 3)

Exp convolution right7 factorial step3 = 6

Exp convolution denominator7 := mul(5040, 6)

Exp convolution denominator7 = 30240

Exp convolution coefficient term7 := div(-19487171/45556992, 30240)

Exp convolution coefficient term7 = -19487171/1377643438080

(11/12)^8

Exp convolution x power8 = 214358881/429981696

(-12/13)^2

Exp convolution y power2 = 144/169

Exp convolution numerator8 := mul(214358881/429981696, 144/169)

Exp convolution numerator8 = 214358881/504631296

Exp convolution left8 factorial step1 := mul(1, 1)

Exp convolution left8 factorial step1 = 1

Exp convolution left8 factorial step2 := mul(1, 2)

Exp convolution left8 factorial step2 = 2

Exp convolution left8 factorial step3 := mul(2, 3)

Exp convolution left8 factorial step3 = 6

Exp convolution left8 factorial step4 := mul(6, 4)

Exp convolution left8 factorial step4 = 24

Exp convolution left8 factorial step5 := mul(24, 5)

Exp convolution left8 factorial step5 = 120

Exp convolution left8 factorial step6 := mul(120, 6)

Exp convolution left8 factorial step6 = 720

Exp convolution left8 factorial step7 := mul(720, 7)

Exp convolution left8 factorial step7 = 5040

Exp convolution left8 factorial step8 := mul(5040, 8)

Exp convolution left8 factorial step8 = 40320

Exp convolution right8 factorial step1 := mul(1, 1)

Exp convolution right8 factorial step1 = 1

Exp convolution right8 factorial step2 := mul(1, 2)

Exp convolution right8 factorial step2 = 2

Exp convolution denominator8 := mul(40320, 2)

Exp convolution denominator8 = 80640

Exp convolution coefficient term8 := div(214358881/504631296, 80640)

Exp convolution coefficient term8 = 214358881/40693467709440

(11/12)^9

Exp convolution x power9 = 2357947691/5159780352

(-12/13)^1

Exp convolution y power1 = -12/13

Exp convolution numerator9 := mul(2357947691/5159780352, -12/13)

Exp convolution numerator9 = -2357947691/5589762048

Exp convolution left9 factorial step1 := mul(1, 1)

Exp convolution left9 factorial step1 = 1

Exp convolution left9 factorial step2 := mul(1, 2)

Exp convolution left9 factorial step2 = 2

Exp convolution left9 factorial step3 := mul(2, 3)

Exp convolution left9 factorial step3 = 6

Exp convolution left9 factorial step4 := mul(6, 4)

Exp convolution left9 factorial step4 = 24

Exp convolution left9 factorial step5 := mul(24, 5)

Exp convolution left9 factorial step5 = 120

Exp convolution left9 factorial step6 := mul(120, 6)

Exp convolution left9 factorial step6 = 720

Exp convolution left9 factorial step7 := mul(720, 7)

Exp convolution left9 factorial step7 = 5040

Exp convolution left9 factorial step8 := mul(5040, 8)

Exp convolution left9 factorial step8 = 40320

Exp convolution left9 factorial step9 := mul(40320, 9)

Exp convolution left9 factorial step9 = 362880

Exp convolution right9 factorial step1 := mul(1, 1)

Exp convolution right9 factorial step1 = 1

Exp convolution denominator9 := mul(362880, 1)

Exp convolution denominator9 = 362880

Exp convolution coefficient term9 := div(-2357947691/5589762048, 362880)

Exp convolution coefficient term9 = -2357947691/2028412851978240

(11/12)^10

Exp convolution x power10 = 25937424601/61917364224

(-12/13)^0

Exp convolution y power0 = 1

Exp convolution numerator10 := mul(25937424601/61917364224, 1)

Exp convolution numerator10 = 25937424601/61917364224

Exp convolution left10 factorial step1 := mul(1, 1)

Exp convolution left10 factorial step1 = 1

Exp convolution left10 factorial step2 := mul(1, 2)

Exp convolution left10 factorial step2 = 2

Exp convolution left10 factorial step3 := mul(2, 3)

Exp convolution left10 factorial step3 = 6

Exp convolution left10 factorial step4 := mul(6, 4)

Exp convolution left10 factorial step4 = 24

Exp convolution left10 factorial step5 := mul(24, 5)

Exp convolution left10 factorial step5 = 120

Exp convolution left10 factorial step6 := mul(120, 6)

Exp convolution left10 factorial step6 = 720

Exp convolution left10 factorial step7 := mul(720, 7)

Exp convolution left10 factorial step7 = 5040

Exp convolution left10 factorial step8 := mul(5040, 8)

Exp convolution left10 factorial step8 = 40320

Exp convolution left10 factorial step9 := mul(40320, 9)

Exp convolution left10 factorial step9 = 362880

Exp convolution left10 factorial step10 := mul(362880, 10)

Exp convolution left10 factorial step10 = 3628800

source roles = [0]

derived = [1]

Exp convolution denominator10 := mul(3628800, 1)

Exp convolution denominator10 = 3628800

Exp convolution coefficient term10 := div(25937424601/61917364224, 3628800)

Exp convolution coefficient term10 = 25937424601/224685731296051200

Exp complete declared-degree convolution := sum(2985984/24125236073575, -456192/371157478055, 156816/28550575235, -31944/2196198095, 14641/579217080, -161051/5346619200, 1771561/71068907520, -19487171/1377643438080, 214358881/40693467709440, -2357947691/2028412851978240, 25937424601/224685731296051200)

Exp complete declared-degree convolution = 1/30974836056463278561086668800

Exp ORIGINAL arguments sum := sum(11/12, -12/13)

Exp ORIGINAL arguments sum = -1/156

(-1/156)^10

Exp sum argument declared power = 1/8535834451185868210176

Exp sum argument factorial step1 := mul(1, 1)

Exp sum argument factorial step1 = 1

Exp sum argument factorial step2 := mul(1, 2)

Exp sum argument factorial step2 = 2

Exp sum argument factorial step3 := mul(2, 3)

Exp sum argument factorial step3 = 6

Exp sum argument factorial step4 := mul(6, 4)

Exp sum argument factorial step4 = 24

Exp sum argument factorial step5 := mul(24, 5)

Exp sum argument factorial step5 = 120

Exp sum argument factorial step6 := mul(120, 6)

Exp sum argument factorial step6 = 720

Exp sum argument factorial step7 := mul(720, 7)

Exp sum argument factorial step7 = 5040

Exp sum argument factorial step8 := mul(5040, 8)

Exp sum argument factorial step8 = 40320

Exp sum argument factorial step9 := mul(40320, 9)

Exp sum argument factorial step9 = 362880

Exp sum argument factorial step10 := mul(362880, 10)

Exp sum argument factorial step10 = 3628800

Exp sum argument coefficient := div(1/8535834451185868210176, 3628800)

Exp sum argument coefficient = 1/30974836056463278561086668800

source roles = [[11/12, -12/13], a_n=1/n! for EVERY n>=0]

derived = [coefficient ratio=1/(n+1), for EVERY fixed real z,term absolute ratio=|z|/(n+1)->0, absolute convergence and real-valued E(z)]

source roles = [[11/12, -12/13, 10], E(x),E(y) absolutely convergent]

derived = [for EVERY n,sum_k x^k*y^(n-k)/(k!*(n-k)!)=(x+y)^n/n!, E(x)E(y)=E(x+y)]

逐步反向核對

• S184 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S183 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S182 · premise_and_result_audit — 以精確有理數重算 Exp sum argument coefficient，不以小數近似替代。
• S181 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step10，不以小數近似替代。
• S180 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step9，不以小數近似替代。
• S179 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step8，不以小數近似替代。
• S178 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step7，不以小數近似替代。
• S177 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step6，不以小數近似替代。
• S176 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step5，不以小數近似替代。
• S175 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step4，不以小數近似替代。
• S174 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step3，不以小數近似替代。
• S173 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step2，不以小數近似替代。
• S172 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step1，不以小數近似替代。
• S171 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S170 · premise_and_result_audit — 以精確有理數重算 Exp ORIGINAL arguments sum，不以小數近似替代。
• S169 · premise_and_result_audit — 以精確有理數重算 Exp complete declared-degree convolution，不以小數近似替代。
• S168 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term10，不以小數近似替代。
• S167 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator10，不以小數近似替代。
• S166 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S165 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step10，不以小數近似替代。
• S164 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step9，不以小數近似替代。
• S163 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step8，不以小數近似替代。
• S162 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step7，不以小數近似替代。
• S161 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step6，不以小數近似替代。
• S160 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step5，不以小數近似替代。
• S159 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step4，不以小數近似替代。
• S158 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step3，不以小數近似替代。
• S157 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step2，不以小數近似替代。
• S156 · premise_and_result_audit — 以精確有理數重算 Exp convolution left10 factorial step1，不以小數近似替代。
• S155 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator10，不以小數近似替代。
• S154 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S153 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S152 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term9，不以小數近似替代。
• S151 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator9，不以小數近似替代。
• S150 · premise_and_result_audit — 以精確有理數重算 Exp convolution right9 factorial step1，不以小數近似替代。
• S149 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step9，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step8，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step7，不以小數近似替代。
• S146 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step6，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step5，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step4，不以小數近似替代。
• S143 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step3，不以小數近似替代。
• S142 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step2，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 Exp convolution left9 factorial step1，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator9，不以小數近似替代。
• S139 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S138 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S137 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term8，不以小數近似替代。
• S136 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator8，不以小數近似替代。
• S135 · premise_and_result_audit — 以精確有理數重算 Exp convolution right8 factorial step2，不以小數近似替代。
• S134 · premise_and_result_audit — 以精確有理數重算 Exp convolution right8 factorial step1，不以小數近似替代。
• S133 · premise_and_result_audit — 以精確有理數重算 Exp convolution left8 factorial step8，不以小數近似替代。
• S132 · premise_and_result_audit — 以精確有理數重算 Exp convolution left8 factorial step7，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 Exp convolution left8 factorial step6，不以小數近似替代。
• S130 · premise_and_result_audit — 以精確有理數重算 Exp convolution left8 factorial step5，不以小數近似替代。
• S129 · premise_and_result_audit — 以精確有理數重算 Exp convolution left8 factorial step4，不以小數近似替代。
• S128 · premise_and_result_audit — 以精確有理數重算 Exp convolution left8 factorial step3，不以小數近似替代。
• S127 · premise_and_result_audit — 以精確有理數重算 Exp convolution left8 factorial step2，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 Exp convolution left8 factorial step1，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator8，不以小數近似替代。
• S124 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S123 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S122 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term7，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator7，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Exp convolution right7 factorial step3，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 Exp convolution right7 factorial step2，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 Exp convolution right7 factorial step1，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 Exp convolution left7 factorial step7，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Exp convolution left7 factorial step6，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Exp convolution left7 factorial step5，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 Exp convolution left7 factorial step4，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Exp convolution left7 factorial step3，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Exp convolution left7 factorial step2，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 Exp convolution left7 factorial step1，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator7，不以小數近似替代。
• S109 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S108 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S107 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term6，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator6，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 Exp convolution right6 factorial step4，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 Exp convolution right6 factorial step3，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Exp convolution right6 factorial step2，不以小數近似替代。
• S102 · premise_and_result_audit — 以精確有理數重算 Exp convolution right6 factorial step1，不以小數近似替代。
• S101 · premise_and_result_audit — 以精確有理數重算 Exp convolution left6 factorial step6，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 Exp convolution left6 factorial step5，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 Exp convolution left6 factorial step4，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Exp convolution left6 factorial step3，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Exp convolution left6 factorial step2，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Exp convolution left6 factorial step1，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator6，不以小數近似替代。
• S094 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S093 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S092 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term5，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator5，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Exp convolution right5 factorial step5，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Exp convolution right5 factorial step4，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 Exp convolution right5 factorial step3，不以小數近似替代。
• S087 · premise_and_result_audit — 以精確有理數重算 Exp convolution right5 factorial step2，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 Exp convolution right5 factorial step1，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Exp convolution left5 factorial step5，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Exp convolution left5 factorial step4，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Exp convolution left5 factorial step3，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Exp convolution left5 factorial step2，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Exp convolution left5 factorial step1，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator5，不以小數近似替代。
• S079 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S078 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S077 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term4，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator4，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 Exp convolution right4 factorial step6，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 Exp convolution right4 factorial step5，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Exp convolution right4 factorial step4，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Exp convolution right4 factorial step3，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Exp convolution right4 factorial step2，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Exp convolution right4 factorial step1，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Exp convolution left4 factorial step4，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Exp convolution left4 factorial step3，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Exp convolution left4 factorial step2，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Exp convolution left4 factorial step1，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator4，不以小數近似替代。
• S064 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S063 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S062 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term3，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator3，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Exp convolution right3 factorial step7，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Exp convolution right3 factorial step6，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Exp convolution right3 factorial step5，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Exp convolution right3 factorial step4，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Exp convolution right3 factorial step3，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Exp convolution right3 factorial step2，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Exp convolution right3 factorial step1，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Exp convolution left3 factorial step3，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Exp convolution left3 factorial step2，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Exp convolution left3 factorial step1，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator3，不以小數近似替代。
• S049 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S047 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term2，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator2，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step8，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step7，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step6，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step5，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step4，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step3，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step2，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step1，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Exp convolution left2 factorial step2，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Exp convolution left2 factorial step1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator2，不以小數近似替代。
• S034 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S033 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S032 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term1，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator1，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step9，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step8，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step7，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step6，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step5，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step4，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step3，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step2，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step1，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Exp convolution left1 factorial step1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator1，不以小數近似替代。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term0，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator0，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step10，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step9，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step8，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step7，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step6，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step5，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step4，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step3，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step2，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step1，不以小數近似替代。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator0，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(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, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。