Question 24210: One Convolution Degree Is Not the Whole Exponential Addition Proof

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-real-exponential-function

Step-by-step derivation

source roles = [exp-addition-convolution]

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

(10/11)^0

Exp convolution x power0 = 1

(-11/12)^9

Exp convolution y power9 = -2357947691/5159780352

Exp convolution numerator0 := mul(1, -2357947691/5159780352)

Exp convolution numerator0 = -2357947691/5159780352

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 denominator0 := mul(1, 362880)

Exp convolution denominator0 = 362880

Exp convolution coefficient term0 := div(-2357947691/5159780352, 362880)

Exp convolution coefficient term0 = -2357947691/1872381094133760

(10/11)^1

Exp convolution x power1 = 10/11

(-11/12)^8

Exp convolution y power8 = 214358881/429981696

Exp convolution numerator1 := mul(10/11, 214358881/429981696)

Exp convolution numerator1 = 97435855/214990848

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 denominator1 := mul(1, 40320)

Exp convolution denominator1 = 40320

Exp convolution coefficient term1 := div(97435855/214990848, 40320)

Exp convolution coefficient term1 = 19487171/1733686198272

(10/11)^2

Exp convolution x power2 = 100/121

(-11/12)^7

Exp convolution y power7 = -19487171/35831808

Exp convolution numerator2 := mul(100/121, -19487171/35831808)

Exp convolution numerator2 = -4026275/8957952

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 denominator2 := mul(2, 5040)

Exp convolution denominator2 = 10080

Exp convolution coefficient term2 := div(-4026275/8957952, 10080)

Exp convolution coefficient term2 = -805255/18059231232

(10/11)^3

Exp convolution x power3 = 1000/1331

(-11/12)^6

Exp convolution y power6 = 1771561/2985984

Exp convolution numerator3 := mul(1000/1331, 1771561/2985984)

Exp convolution numerator3 = 166375/373248

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 denominator3 := mul(6, 720)

Exp convolution denominator3 = 4320

Exp convolution coefficient term3 := div(166375/373248, 4320)

Exp convolution coefficient term3 = 33275/322486272

(10/11)^4

Exp convolution x power4 = 10000/14641

(-11/12)^5

Exp convolution y power5 = -161051/248832

Exp convolution numerator4 := mul(10000/14641, -161051/248832)

Exp convolution numerator4 = -6875/15552

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 denominator4 := mul(24, 120)

Exp convolution denominator4 = 2880

Exp convolution coefficient term4 := div(-6875/15552, 2880)

Exp convolution coefficient term4 = -1375/8957952

(10/11)^5

Exp convolution x power5 = 100000/161051

(-11/12)^4

Exp convolution y power4 = 14641/20736

Exp convolution numerator5 := mul(100000/161051, 14641/20736)

Exp convolution numerator5 = 3125/7128

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 denominator5 := mul(120, 24)

Exp convolution denominator5 = 2880

Exp convolution coefficient term5 := div(3125/7128, 2880)

Exp convolution coefficient term5 = 625/4105728

(10/11)^6

Exp convolution x power6 = 1000000/1771561

(-11/12)^3

Exp convolution y power3 = -1331/1728

Exp convolution numerator6 := mul(1000000/1771561, -1331/1728)

Exp convolution numerator6 = -15625/35937

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 denominator6 := mul(720, 6)

Exp convolution denominator6 = 4320

Exp convolution coefficient term6 := div(-15625/35937, 4320)

Exp convolution coefficient term6 = -3125/31049568

(10/11)^7

Exp convolution x power7 = 10000000/19487171

(-11/12)^2

Exp convolution y power2 = 121/144

Exp convolution numerator7 := mul(10000000/19487171, 121/144)

Exp convolution numerator7 = 625000/1449459

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 denominator7 := mul(5040, 2)

Exp convolution denominator7 = 10080

Exp convolution coefficient term7 := div(625000/1449459, 10080)

Exp convolution coefficient term7 = 15625/365263668

(10/11)^8

Exp convolution x power8 = 100000000/214358881

(-11/12)^1

Exp convolution y power1 = -11/12

Exp convolution numerator8 := mul(100000000/214358881, -11/12)

Exp convolution numerator8 = -25000000/58461513

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 denominator8 := mul(40320, 1)

Exp convolution denominator8 = 40320

Exp convolution coefficient term8 := div(-25000000/58461513, 40320)

Exp convolution coefficient term8 = -78125/7366150638

(10/11)^9

Exp convolution x power9 = 1000000000/2357947691

(-11/12)^0

Exp convolution y power0 = 1

Exp convolution numerator9 := mul(1000000000/2357947691, 1)

Exp convolution numerator9 = 1000000000/2357947691

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

source roles = [0]

derived = [1]

Exp convolution denominator9 := mul(362880, 1)

Exp convolution denominator9 = 362880

Exp convolution coefficient term9 := div(1000000000/2357947691, 362880)

Exp convolution coefficient term9 = 1562500/1336956340797

Exp complete declared-degree convolution := sum(-2357947691/1872381094133760, 19487171/1733686198272, -805255/18059231232, 33275/322486272, -1375/8957952, 625/4105728, -3125/31049568, 15625/365263668, -78125/7366150638, 1562500/1336956340797)

Exp complete declared-degree convolution = -1/4414976677584753037148160

Exp ORIGINAL arguments sum := sum(10/11, -11/12)

Exp ORIGINAL arguments sum = -1/132

(-1/132)^9

Exp sum argument declared power = -1/12166492167065567232

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 coefficient := div(-1/12166492167065567232, 362880)

Exp sum argument coefficient = -1/4414976677584753037148160

source roles = [[10/11, -11/12], 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 = [[10/11, -11/12, 9], 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)]

Reverse checks

• S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S156 · premise_and_result_audit — Recompute Exp sum argument coefficient with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Exp sum argument factorial step9 with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Exp sum argument factorial step8 with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute Exp sum argument factorial step7 with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute Exp sum argument factorial step6 with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Exp sum argument factorial step5 with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute Exp sum argument factorial step4 with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute Exp sum argument factorial step3 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Exp sum argument factorial step2 with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute Exp sum argument factorial step1 with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Recompute Exp ORIGINAL arguments sum with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute Exp complete declared-degree convolution with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute Exp convolution coefficient term9 with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute Exp convolution denominator9 with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Recompute Exp convolution left9 factorial step9 with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Exp convolution left9 factorial step8 with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Exp convolution left9 factorial step7 with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Exp convolution left9 factorial step6 with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Exp convolution left9 factorial step5 with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Exp convolution left9 factorial step4 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Exp convolution left9 factorial step3 with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Exp convolution left9 factorial step2 with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Exp convolution left9 factorial step1 with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Exp convolution numerator9 with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Recompute Exp convolution coefficient term8 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Exp convolution denominator8 with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Exp convolution right8 factorial step1 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Exp convolution left8 factorial step8 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Exp convolution left8 factorial step7 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute Exp convolution left8 factorial step6 with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute Exp convolution left8 factorial step5 with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Exp convolution left8 factorial step4 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Exp convolution left8 factorial step3 with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Exp convolution left8 factorial step2 with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Exp convolution left8 factorial step1 with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Exp convolution numerator8 with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Recompute Exp convolution coefficient term7 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Exp convolution denominator7 with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Exp convolution right7 factorial step2 with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Exp convolution right7 factorial step1 with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Exp convolution left7 factorial step7 with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Exp convolution left7 factorial step6 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute Exp convolution left7 factorial step5 with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute Exp convolution left7 factorial step4 with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Exp convolution left7 factorial step3 with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Exp convolution left7 factorial step2 with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Exp convolution left7 factorial step1 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute Exp convolution numerator7 with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Recompute Exp convolution coefficient term6 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Exp convolution denominator6 with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Exp convolution right6 factorial step3 with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Exp convolution right6 factorial step2 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Exp convolution right6 factorial step1 with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Exp convolution left6 factorial step6 with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Exp convolution left6 factorial step5 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Exp convolution left6 factorial step4 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Exp convolution left6 factorial step3 with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Exp convolution left6 factorial step2 with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Exp convolution left6 factorial step1 with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Exp convolution numerator6 with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Recompute Exp convolution coefficient term5 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Exp convolution denominator5 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Exp convolution right5 factorial step4 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Exp convolution right5 factorial step3 with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Exp convolution right5 factorial step2 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Exp convolution right5 factorial step1 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Exp convolution left5 factorial step5 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Exp convolution left5 factorial step4 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Exp convolution left5 factorial step3 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Exp convolution left5 factorial step2 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Exp convolution left5 factorial step1 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Exp convolution numerator5 with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Recompute Exp convolution coefficient term4 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Exp convolution denominator4 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Exp convolution right4 factorial step5 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Exp convolution right4 factorial step4 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Exp convolution right4 factorial step3 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Exp convolution right4 factorial step2 with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Exp convolution right4 factorial step1 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Exp convolution left4 factorial step4 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Exp convolution left4 factorial step3 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Exp convolution left4 factorial step2 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Exp convolution left4 factorial step1 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Exp convolution numerator4 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Recompute Exp convolution coefficient term3 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Exp convolution denominator3 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Exp convolution right3 factorial step6 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Exp convolution right3 factorial step5 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Exp convolution right3 factorial step4 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Exp convolution right3 factorial step3 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Exp convolution right3 factorial step2 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Exp convolution right3 factorial step1 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Exp convolution left3 factorial step3 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Exp convolution left3 factorial step2 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Exp convolution left3 factorial step1 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Exp convolution numerator3 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Recompute Exp convolution coefficient term2 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Exp convolution denominator2 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Exp convolution right2 factorial step7 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Exp convolution right2 factorial step6 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Exp convolution right2 factorial step5 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Exp convolution right2 factorial step4 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Exp convolution right2 factorial step3 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Exp convolution right2 factorial step2 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Exp convolution right2 factorial step1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Exp convolution left2 factorial step2 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Exp convolution left2 factorial step1 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Exp convolution numerator2 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute Exp convolution coefficient term1 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Exp convolution denominator1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Exp convolution right1 factorial step8 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Exp convolution right1 factorial step7 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Exp convolution right1 factorial step6 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Exp convolution right1 factorial step5 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Exp convolution right1 factorial step4 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Exp convolution right1 factorial step3 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Exp convolution right1 factorial step2 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Exp convolution right1 factorial step1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Exp convolution left1 factorial step1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Exp convolution numerator1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute Exp convolution coefficient term0 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Exp convolution denominator0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Exp convolution right0 factorial step9 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Exp convolution right0 factorial step8 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Exp convolution right0 factorial step7 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Exp convolution right0 factorial step6 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Exp convolution right0 factorial step5 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Exp convolution right0 factorial step4 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Exp convolution right0 factorial step3 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Exp convolution right0 factorial step2 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Exp convolution right0 factorial step1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute Exp convolution numerator0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(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)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.