第 24212 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":[12,13],"y":[-13,14],"degree":11}。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":[12,13],"y":[-13,14],"degree":11}。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]]

(12/13)^0

Exp convolution x power0 = 1

(-13/14)^11

Exp convolution y power11 = -1792160394037/4049565169664

Exp convolution numerator0 := mul(1, -1792160394037/4049565169664)

Exp convolution numerator0 = -1792160394037/4049565169664

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 right0 factorial step11 := mul(3628800, 11)

Exp convolution right0 factorial step11 = 39916800

Exp convolution denominator0 := mul(1, 39916800)

Exp convolution denominator0 = 39916800

Exp convolution coefficient term0 := div(-1792160394037/4049565169664, 39916800)

Exp convolution coefficient term0 = -1792160394037/161645682964443955200

(12/13)^1

Exp convolution x power1 = 12/13

(-13/14)^10

Exp convolution y power10 = 137858491849/289254654976

Exp convolution numerator1 := mul(12/13, 137858491849/289254654976)

Exp convolution numerator1 = 31813498119/72313663744

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 right1 factorial step10 := mul(362880, 10)

Exp convolution right1 factorial step10 = 3628800

Exp convolution denominator1 := mul(1, 3628800)

Exp convolution denominator1 = 3628800

Exp convolution coefficient term1 := div(31813498119/72313663744, 3628800)

Exp convolution coefficient term1 = 10604499373/87470607664742400

(12/13)^2

Exp convolution x power2 = 144/169

(-13/14)^9

Exp convolution y power9 = -10604499373/20661046784

Exp convolution numerator2 := mul(144/169, -10604499373/20661046784)

Exp convolution numerator2 = -564736653/1291315424

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 right2 factorial step9 := mul(40320, 9)

Exp convolution right2 factorial step9 = 362880

Exp convolution denominator2 := mul(2, 362880)

Exp convolution denominator2 = 725760

Exp convolution coefficient term2 := div(-564736653/1291315424, 725760)

Exp convolution coefficient term2 = -62748517/104131675791360

(12/13)^3

Exp convolution x power3 = 1728/2197

(-13/14)^8

Exp convolution y power8 = 815730721/1475789056

Exp convolution numerator3 := mul(1728/2197, 815730721/1475789056)

Exp convolution numerator3 = 10024911/23059204

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 right3 factorial step8 := mul(5040, 8)

Exp convolution right3 factorial step8 = 40320

Exp convolution denominator3 := mul(6, 40320)

Exp convolution denominator3 = 241920

Exp convolution coefficient term3 := div(10024911/23059204, 241920)

Exp convolution coefficient term3 = 371293/206610467840

(12/13)^4

Exp convolution x power4 = 20736/28561

(-13/14)^7

Exp convolution y power7 = -62748517/105413504

Exp convolution numerator4 := mul(20736/28561, -62748517/105413504)

Exp convolution numerator4 = -355914/823543

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 right4 factorial step7 := mul(720, 7)

Exp convolution right4 factorial step7 = 5040

Exp convolution denominator4 := mul(24, 5040)

Exp convolution denominator4 = 120960

Exp convolution coefficient term4 := div(-355914/823543, 120960)

Exp convolution coefficient term4 = -6591/1844736320

(12/13)^5

Exp convolution x power5 = 248832/371293

(-13/14)^6

Exp convolution y power6 = 4826809/7529536

Exp convolution numerator5 := mul(248832/371293, 4826809/7529536)

Exp convolution numerator5 = 50544/117649

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 right5 factorial step6 := mul(120, 6)

Exp convolution right5 factorial step6 = 720

Exp convolution denominator5 := mul(120, 720)

Exp convolution denominator5 = 86400

Exp convolution coefficient term5 := div(50544/117649, 86400)

Exp convolution coefficient term5 = 117/23529800

(12/13)^6

Exp convolution x power6 = 2985984/4826809

(-13/14)^5

Exp convolution y power5 = -371293/537824

Exp convolution numerator6 := mul(2985984/4826809, -371293/537824)

Exp convolution numerator6 = -93312/218491

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 right6 factorial step5 := mul(24, 5)

Exp convolution right6 factorial step5 = 120

Exp convolution denominator6 := mul(720, 120)

Exp convolution denominator6 = 86400

Exp convolution coefficient term6 := div(-93312/218491, 86400)

Exp convolution coefficient term6 = -27/5462275

(12/13)^7

Exp convolution x power7 = 35831808/62748517

(-13/14)^4

Exp convolution y power4 = 28561/38416

Exp convolution numerator7 := mul(35831808/62748517, 28561/38416)

Exp convolution numerator7 = 2239488/5274997

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 right7 factorial step4 := mul(6, 4)

Exp convolution right7 factorial step4 = 24

Exp convolution denominator7 := mul(5040, 24)

Exp convolution denominator7 = 120960

Exp convolution coefficient term7 := div(2239488/5274997, 120960)

Exp convolution coefficient term7 = 648/184624895

(12/13)^8

Exp convolution x power8 = 429981696/815730721

(-13/14)^3

Exp convolution y power3 = -2197/2744

Exp convolution numerator8 := mul(429981696/815730721, -2197/2744)

Exp convolution numerator8 = -53747712/127353499

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 right8 factorial step3 := mul(2, 3)

Exp convolution right8 factorial step3 = 6

Exp convolution denominator8 := mul(40320, 6)

Exp convolution denominator8 = 241920

Exp convolution coefficient term8 := div(-53747712/127353499, 241920)

Exp convolution coefficient term8 = -7776/4457372465

(12/13)^9

Exp convolution x power9 = 5159780352/10604499373

(-13/14)^2

Exp convolution y power2 = 169/196

Exp convolution numerator9 := mul(5159780352/10604499373, 169/196)

Exp convolution numerator9 = 1289945088/3074677333

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 right9 factorial step2 := mul(1, 2)

Exp convolution right9 factorial step2 = 2

Exp convolution denominator9 := mul(362880, 2)

Exp convolution denominator9 = 725760

Exp convolution coefficient term9 := div(1289945088/3074677333, 725760)

Exp convolution coefficient term9 = 62208/107613706655

(12/13)^10

Exp convolution x power10 = 61917364224/137858491849

(-13/14)^1

Exp convolution y power1 = -13/14

Exp convolution numerator10 := mul(61917364224/137858491849, -13/14)

Exp convolution numerator10 = -30958682112/74231495611

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

Exp convolution right10 factorial step1 := mul(1, 1)

Exp convolution right10 factorial step1 = 1

Exp convolution denominator10 := mul(3628800, 1)

Exp convolution denominator10 = 3628800

Exp convolution coefficient term10 := div(-30958682112/74231495611, 3628800)

Exp convolution coefficient term10 = -1492992/12990511731925

(12/13)^11

Exp convolution x power11 = 743008370688/1792160394037

(-13/14)^0

Exp convolution y power0 = 1

Exp convolution numerator11 := mul(743008370688/1792160394037, 1)

Exp convolution numerator11 = 743008370688/1792160394037

Exp convolution left11 factorial step1 := mul(1, 1)

Exp convolution left11 factorial step1 = 1

Exp convolution left11 factorial step2 := mul(1, 2)

Exp convolution left11 factorial step2 = 2

Exp convolution left11 factorial step3 := mul(2, 3)

Exp convolution left11 factorial step3 = 6

Exp convolution left11 factorial step4 := mul(6, 4)

Exp convolution left11 factorial step4 = 24

Exp convolution left11 factorial step5 := mul(24, 5)

Exp convolution left11 factorial step5 = 120

Exp convolution left11 factorial step6 := mul(120, 6)

Exp convolution left11 factorial step6 = 720

Exp convolution left11 factorial step7 := mul(720, 7)

Exp convolution left11 factorial step7 = 5040

Exp convolution left11 factorial step8 := mul(5040, 8)

Exp convolution left11 factorial step8 = 40320

Exp convolution left11 factorial step9 := mul(40320, 9)

Exp convolution left11 factorial step9 = 362880

Exp convolution left11 factorial step10 := mul(362880, 10)

Exp convolution left11 factorial step10 = 3628800

Exp convolution left11 factorial step11 := mul(3628800, 11)

Exp convolution left11 factorial step11 = 39916800

source roles = [0]

derived = [1]

Exp convolution denominator11 := mul(39916800, 1)

Exp convolution denominator11 = 39916800

Exp convolution coefficient term11 := div(743008370688/1792160394037, 39916800)

Exp convolution coefficient term11 = 35831808/3449908758521225

Exp complete declared-degree convolution := sum(-1792160394037/161645682964443955200, 10604499373/87470607664742400, -62748517/104131675791360, 371293/206610467840, -6591/1844736320, 117/23529800, -27/5462275, 648/184624895, -7776/4457372465, 62208/107613706655, -1492992/12990511731925, 35831808/3449908758521225)

Exp complete declared-degree convolution = -1/289694990875937857011834775142400

Exp ORIGINAL arguments sum := sum(12/13, -13/14)

Exp ORIGINAL arguments sum = -1/182

(-1/182)^11

Exp sum argument declared power = -1/7257470310143544998893568

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 factorial step11 := mul(3628800, 11)

Exp sum argument factorial step11 = 39916800

Exp sum argument coefficient := div(-1/7257470310143544998893568, 39916800)

Exp sum argument coefficient = -1/289694990875937857011834775142400

source roles = [[12/13, -13/14], 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 = [[12/13, -13/14, 11], 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

• S212 · 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。
• S211 · 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。
• S210 · premise_and_result_audit — 以精確有理數重算 Exp sum argument coefficient，不以小數近似替代。
• S209 · premise_and_result_audit — 以精確有理數重算 Exp sum argument factorial step11，不以小數近似替代。
• S208 · 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。
• S207 · 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。
• S206 · 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。
• S205 · 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。
• S204 · 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。
• S203 · 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。
• S202 · 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。
• S201 · 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。
• S200 · 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。
• S199 · 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。
• S198 · 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。
• S197 · premise_and_result_audit — 以精確有理數重算 Exp ORIGINAL arguments sum，不以小數近似替代。
• S196 · 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。
• S195 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term11，不以小數近似替代。
• S194 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator11，不以小數近似替代。
• S193 · 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。
• S192 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step11，不以小數近似替代。
• S191 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step10，不以小數近似替代。
• S190 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step9，不以小數近似替代。
• S189 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step8，不以小數近似替代。
• S188 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step7，不以小數近似替代。
• S187 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step6，不以小數近似替代。
• S186 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step5，不以小數近似替代。
• S185 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step4，不以小數近似替代。
• S184 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step3，不以小數近似替代。
• S183 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step2，不以小數近似替代。
• S182 · premise_and_result_audit — 以精確有理數重算 Exp convolution left11 factorial step1，不以小數近似替代。
• S181 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator11，不以小數近似替代。
• S180 · 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。
• S179 · 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。
• S178 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term10，不以小數近似替代。
• S177 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator10，不以小數近似替代。
• S176 · premise_and_result_audit — 以精確有理數重算 Exp convolution right10 factorial step1，不以小數近似替代。
• S175 · 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。
• S174 · 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。
• S173 · 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。
• S172 · 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。
• S171 · 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。
• S170 · 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。
• S169 · 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。
• S168 · 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。
• S167 · 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。
• S166 · 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。
• S165 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator10，不以小數近似替代。
• S164 · 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。
• S163 · 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。
• S162 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term9，不以小數近似替代。
• S161 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator9，不以小數近似替代。
• S160 · premise_and_result_audit — 以精確有理數重算 Exp convolution right9 factorial step2，不以小數近似替代。
• S159 · 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。
• S158 · 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。
• S157 · 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。
• S156 · 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。
• S155 · 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。
• S154 · 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。
• S153 · 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。
• S152 · 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。
• S151 · 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。
• S150 · 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。
• S149 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator9，不以小數近似替代。
• S148 · 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。
• S147 · 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。
• S146 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term8，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator8，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 Exp convolution right8 factorial step3，不以小數近似替代。
• S143 · 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。
• S142 · 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。
• S141 · 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。
• S140 · 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。
• S139 · 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。
• S138 · 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。
• S137 · 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。
• S136 · 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。
• S135 · 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。
• S134 · 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。
• S133 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator8，不以小數近似替代。
• S132 · 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。
• S131 · 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。
• S130 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term7，不以小數近似替代。
• S129 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator7，不以小數近似替代。
• S128 · premise_and_result_audit — 以精確有理數重算 Exp convolution right7 factorial step4，不以小數近似替代。
• S127 · 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。
• S126 · 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。
• S125 · 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。
• S124 · 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。
• S123 · 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。
• S122 · 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。
• S121 · 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。
• S120 · 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。
• S119 · 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。
• S118 · 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。
• S117 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator7，不以小數近似替代。
• S116 · 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。
• S115 · 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。
• S114 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term6，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator6，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Exp convolution right6 factorial step5，不以小數近似替代。
• S111 · 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。
• S110 · 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。
• S109 · 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。
• S108 · 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。
• S107 · 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。
• S106 · 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。
• S105 · 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。
• S104 · 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。
• S103 · 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。
• S102 · 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。
• S101 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator6，不以小數近似替代。
• S100 · 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。
• S099 · 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。
• S098 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term5，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator5，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Exp convolution right5 factorial step6，不以小數近似替代。
• S095 · 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。
• S094 · 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。
• S093 · 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。
• S092 · 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。
• S091 · 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。
• S090 · 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。
• S089 · 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。
• S088 · 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。
• S087 · 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。
• S086 · 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。
• S085 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator5，不以小數近似替代。
• S084 · 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。
• S083 · 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。
• S082 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term4，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator4，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Exp convolution right4 factorial step7，不以小數近似替代。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator4，不以小數近似替代。
• S068 · 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。
• S067 · 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。
• S066 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term3，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator3，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Exp convolution right3 factorial step8，不以小數近似替代。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · 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。
• S055 · 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。
• S054 · 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。
• S053 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator3，不以小數近似替代。
• S052 · 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。
• S051 · 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。
• S050 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term2，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator2，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Exp convolution right2 factorial step9，不以小數近似替代。
• S047 · 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。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• S037 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator2，不以小數近似替代。
• S036 · 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。
• S035 · 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。
• S034 · premise_and_result_audit — 以精確有理數重算 Exp convolution coefficient term1，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator1，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Exp convolution right1 factorial step10，不以小數近似替代。
• S031 · 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。
• S030 · 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。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · 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。
• S024 · 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。
• S023 · 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。
• S022 · 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。
• S021 · premise_and_result_audit — 以精確有理數重算 Exp convolution numerator1，不以小數近似替代。
• S020 · 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。
• 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 — 以精確有理數重算 Exp convolution coefficient term0，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Exp convolution denominator0，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Exp convolution right0 factorial step11，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

(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, 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。