# Question 24209: One Convolution Degree Is Not the Whole Exponential Addition Proof [Return to catalogue page 122](https://ocme.evemisslab.com/en/questions/page/122/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261113-r02-real-exponential-function-foundations/#q-20261015-advance-20261113-r02-real-exponential-function-0059) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0047-24209 · q-20261015-advance-20261113-r02-real-exponential-function-0059 ## 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=9/10, y=-10/11, declared degree=8. 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=9/10, y=-10/11, declared degree=8. 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](https://ocme.evemisslab.com/en/mko/mko-real-exponential-function/) ## Step-by-step derivation S001 · impliesBefore source roles = [exp-addition-convolution] After derived = [[x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate]] Conditions: none List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore (9/10)^0 After Exp convolution x power0 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S001 S003 · impliesBefore (-10/11)^8 After Exp convolution y power8 = 100000000/214358881 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S002 S004 · calculationBefore Exp convolution numerator0 := mul(1, 100000000/214358881) After Exp convolution numerator0 = 100000000/214358881 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [0] After derived = [1] Conditions: none The empty product0! is1,not0. Step identifiers and dependenciesRule: empty_factorial_is_multiplicative_unit · Uses: G001, S004 S006 · calculationBefore Exp convolution right0 factorial step1 := mul(1, 1) After Exp convolution right0 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Exp convolution right0 factorial step2 := mul(1, 2) After Exp convolution right0 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Exp convolution right0 factorial step3 := mul(2, 3) After Exp convolution right0 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Exp convolution right0 factorial step4 := mul(6, 4) After Exp convolution right0 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Exp convolution right0 factorial step5 := mul(24, 5) After Exp convolution right0 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Exp convolution right0 factorial step6 := mul(120, 6) After Exp convolution right0 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Exp convolution right0 factorial step7 := mul(720, 7) After Exp convolution right0 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Exp convolution right0 factorial step8 := mul(5040, 8) After Exp convolution right0 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Exp convolution denominator0 := mul(1, 40320) After Exp convolution denominator0 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Exp convolution coefficient term0 := div(100000000/214358881, 40320) After Exp convolution coefficient term0 = 156250/13504609503 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore (9/10)^1 After Exp convolution x power1 = 9/10 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S015 S017 · impliesBefore (-10/11)^7 After Exp convolution y power7 = -10000000/19487171 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S016 S018 · calculationBefore Exp convolution numerator1 := mul(9/10, -10000000/19487171) After Exp convolution numerator1 = -9000000/19487171 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Exp convolution left1 factorial step1 := mul(1, 1) After Exp convolution left1 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Exp convolution right1 factorial step1 := mul(1, 1) After Exp convolution right1 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Exp convolution right1 factorial step2 := mul(1, 2) After Exp convolution right1 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Exp convolution right1 factorial step3 := mul(2, 3) After Exp convolution right1 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Exp convolution right1 factorial step4 := mul(6, 4) After Exp convolution right1 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Exp convolution right1 factorial step5 := mul(24, 5) After Exp convolution right1 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Exp convolution right1 factorial step6 := mul(120, 6) After Exp convolution right1 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Exp convolution right1 factorial step7 := mul(720, 7) After Exp convolution right1 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Exp convolution denominator1 := mul(1, 5040) After Exp convolution denominator1 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Exp convolution coefficient term1 := div(-9000000/19487171, 5040) After Exp convolution coefficient term1 = -12500/136410197 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore (9/10)^2 After Exp convolution x power2 = 81/100 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S028 S030 · impliesBefore (-10/11)^6 After Exp convolution y power6 = 1000000/1771561 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S029 S031 · calculationBefore Exp convolution numerator2 := mul(81/100, 1000000/1771561) After Exp convolution numerator2 = 810000/1771561 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Exp convolution left2 factorial step1 := mul(1, 1) After Exp convolution left2 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Exp convolution left2 factorial step2 := mul(1, 2) After Exp convolution left2 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Exp convolution right2 factorial step1 := mul(1, 1) After Exp convolution right2 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Exp convolution right2 factorial step2 := mul(1, 2) After Exp convolution right2 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Exp convolution right2 factorial step3 := mul(2, 3) After Exp convolution right2 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Exp convolution right2 factorial step4 := mul(6, 4) After Exp convolution right2 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Exp convolution right2 factorial step5 := mul(24, 5) After Exp convolution right2 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Exp convolution right2 factorial step6 := mul(120, 6) After Exp convolution right2 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Exp convolution denominator2 := mul(2, 720) After Exp convolution denominator2 = 1440 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Exp convolution coefficient term2 := div(810000/1771561, 1440) After Exp convolution coefficient term2 = 1125/3543122 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · impliesBefore (9/10)^3 After Exp convolution x power3 = 729/1000 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S041 S043 · impliesBefore (-10/11)^5 After Exp convolution y power5 = -100000/161051 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S042 S044 · calculationBefore Exp convolution numerator3 := mul(729/1000, -100000/161051) After Exp convolution numerator3 = -72900/161051 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Exp convolution left3 factorial step1 := mul(1, 1) After Exp convolution left3 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Exp convolution left3 factorial step2 := mul(1, 2) After Exp convolution left3 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Exp convolution left3 factorial step3 := mul(2, 3) After Exp convolution left3 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Exp convolution right3 factorial step1 := mul(1, 1) After Exp convolution right3 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore Exp convolution right3 factorial step2 := mul(1, 2) After Exp convolution right3 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Exp convolution right3 factorial step3 := mul(2, 3) After Exp convolution right3 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Exp convolution right3 factorial step4 := mul(6, 4) After Exp convolution right3 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore Exp convolution right3 factorial step5 := mul(24, 5) After Exp convolution right3 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore Exp convolution denominator3 := mul(6, 120) After Exp convolution denominator3 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore Exp convolution coefficient term3 := div(-72900/161051, 720) After Exp convolution coefficient term3 = -405/644204 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · impliesBefore (9/10)^4 After Exp convolution x power4 = 6561/10000 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S054 S056 · impliesBefore (-10/11)^4 After Exp convolution y power4 = 10000/14641 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S055 S057 · calculationBefore Exp convolution numerator4 := mul(6561/10000, 10000/14641) After Exp convolution numerator4 = 6561/14641 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore Exp convolution left4 factorial step1 := mul(1, 1) After Exp convolution left4 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore Exp convolution left4 factorial step2 := mul(1, 2) After Exp convolution left4 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore Exp convolution left4 factorial step3 := mul(2, 3) After Exp convolution left4 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore Exp convolution left4 factorial step4 := mul(6, 4) After Exp convolution left4 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore Exp convolution right4 factorial step1 := mul(1, 1) After Exp convolution right4 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore Exp convolution right4 factorial step2 := mul(1, 2) After Exp convolution right4 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · calculationBefore Exp convolution right4 factorial step3 := mul(2, 3) After Exp convolution right4 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore Exp convolution right4 factorial step4 := mul(6, 4) After Exp convolution right4 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore Exp convolution denominator4 := mul(24, 24) After Exp convolution denominator4 = 576 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore Exp convolution coefficient term4 := div(6561/14641, 576) After Exp convolution coefficient term4 = 729/937024 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · impliesBefore (9/10)^5 After Exp convolution x power5 = 59049/100000 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S067 S069 · impliesBefore (-10/11)^3 After Exp convolution y power3 = -1000/1331 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S068 S070 · calculationBefore Exp convolution numerator5 := mul(59049/100000, -1000/1331) After Exp convolution numerator5 = -59049/133100 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore Exp convolution left5 factorial step1 := mul(1, 1) After Exp convolution left5 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore Exp convolution left5 factorial step2 := mul(1, 2) After Exp convolution left5 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore Exp convolution left5 factorial step3 := mul(2, 3) After Exp convolution left5 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore Exp convolution left5 factorial step4 := mul(6, 4) After Exp convolution left5 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · calculationBefore Exp convolution left5 factorial step5 := mul(24, 5) After Exp convolution left5 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore Exp convolution right5 factorial step1 := mul(1, 1) After Exp convolution right5 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore Exp convolution right5 factorial step2 := mul(1, 2) After Exp convolution right5 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore Exp convolution right5 factorial step3 := mul(2, 3) After Exp convolution right5 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · calculationBefore Exp convolution denominator5 := mul(120, 6) After Exp convolution denominator5 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · calculationBefore Exp convolution coefficient term5 := div(-59049/133100, 720) After Exp convolution coefficient term5 = -6561/10648000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · impliesBefore (9/10)^6 After Exp convolution x power6 = 531441/1000000 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S080 S082 · impliesBefore (-10/11)^2 After Exp convolution y power2 = 100/121 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S081 S083 · calculationBefore Exp convolution numerator6 := mul(531441/1000000, 100/121) After Exp convolution numerator6 = 531441/1210000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore Exp convolution left6 factorial step1 := mul(1, 1) After Exp convolution left6 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore Exp convolution left6 factorial step2 := mul(1, 2) After Exp convolution left6 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · calculationBefore Exp convolution left6 factorial step3 := mul(2, 3) After Exp convolution left6 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085 S087 · calculationBefore Exp convolution left6 factorial step4 := mul(6, 4) After Exp convolution left6 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · calculationBefore Exp convolution left6 factorial step5 := mul(24, 5) After Exp convolution left6 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · calculationBefore Exp convolution left6 factorial step6 := mul(120, 6) After Exp convolution left6 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · calculationBefore Exp convolution right6 factorial step1 := mul(1, 1) After Exp convolution right6 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089 S091 · calculationBefore Exp convolution right6 factorial step2 := mul(1, 2) After Exp convolution right6 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S090 S092 · calculationBefore Exp convolution denominator6 := mul(720, 2) After Exp convolution denominator6 = 1440 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · calculationBefore Exp convolution coefficient term6 := div(531441/1210000, 1440) After Exp convolution coefficient term6 = 59049/193600000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S092 S094 · impliesBefore (9/10)^7 After Exp convolution x power7 = 4782969/10000000 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S093 S095 · impliesBefore (-10/11)^1 After Exp convolution y power1 = -10/11 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S094 S096 · calculationBefore Exp convolution numerator7 := mul(4782969/10000000, -10/11) After Exp convolution numerator7 = -4782969/11000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S095 S097 · calculationBefore Exp convolution left7 factorial step1 := mul(1, 1) After Exp convolution left7 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · calculationBefore Exp convolution left7 factorial step2 := mul(1, 2) After Exp convolution left7 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097 S099 · calculationBefore Exp convolution left7 factorial step3 := mul(2, 3) After Exp convolution left7 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S098 S100 · calculationBefore Exp convolution left7 factorial step4 := mul(6, 4) After Exp convolution left7 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099 S101 · calculationBefore Exp convolution left7 factorial step5 := mul(24, 5) After Exp convolution left7 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S100 S102 · calculationBefore Exp convolution left7 factorial step6 := mul(120, 6) After Exp convolution left7 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · calculationBefore Exp convolution left7 factorial step7 := mul(720, 7) After Exp convolution left7 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · calculationBefore Exp convolution right7 factorial step1 := mul(1, 1) After Exp convolution right7 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · calculationBefore Exp convolution denominator7 := mul(5040, 1) After Exp convolution denominator7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S104 S106 · calculationBefore Exp convolution coefficient term7 := div(-4782969/11000000, 5040) After Exp convolution coefficient term7 = -531441/6160000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105 S107 · impliesBefore (9/10)^8 After Exp convolution x power8 = 43046721/100000000 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S106 S108 · impliesBefore (-10/11)^0 After Exp convolution y power0 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S107 S109 · calculationBefore Exp convolution numerator8 := mul(43046721/100000000, 1) After Exp convolution numerator8 = 43046721/100000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S108 S110 · calculationBefore Exp convolution left8 factorial step1 := mul(1, 1) After Exp convolution left8 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S109 S111 · calculationBefore Exp convolution left8 factorial step2 := mul(1, 2) After Exp convolution left8 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S110 S112 · calculationBefore Exp convolution left8 factorial step3 := mul(2, 3) After Exp convolution left8 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111 S113 · calculationBefore Exp convolution left8 factorial step4 := mul(6, 4) After Exp convolution left8 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S112 S114 · calculationBefore Exp convolution left8 factorial step5 := mul(24, 5) After Exp convolution left8 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S113 S115 · calculationBefore Exp convolution left8 factorial step6 := mul(120, 6) After Exp convolution left8 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S114 S116 · calculationBefore Exp convolution left8 factorial step7 := mul(720, 7) After Exp convolution left8 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S115 S117 · calculationBefore Exp convolution left8 factorial step8 := mul(5040, 8) After Exp convolution left8 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S116 S118 · impliesBefore source roles = [0] After derived = [1] Conditions: none The empty product0! is1,not0. Step identifiers and dependenciesRule: empty_factorial_is_multiplicative_unit · Uses: G001, S117 S119 · calculationBefore Exp convolution denominator8 := mul(40320, 1) After Exp convolution denominator8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S118 S120 · calculationBefore Exp convolution coefficient term8 := div(43046721/100000000, 40320) After Exp convolution coefficient term8 = 4782969/448000000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119 S121 · calculationBefore Exp complete declared-degree convolution := sum(156250/13504609503, -12500/136410197, 1125/3543122, -405/644204, 729/937024, -6561/10648000, 59049/193600000, -531441/6160000000, 4782969/448000000000) After Exp complete declared-degree convolution = 1/864295008192000000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S120 S122 · calculationBefore Exp ORIGINAL arguments sum := sum(9/10, -10/11) After Exp ORIGINAL arguments sum = -1/110 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S121 S123 · impliesBefore (-1/110)^8 After Exp sum argument declared power = 1/21435888100000000 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S122 S124 · calculationBefore Exp sum argument factorial step1 := mul(1, 1) After Exp sum argument factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123 S125 · calculationBefore Exp sum argument factorial step2 := mul(1, 2) After Exp sum argument factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124 S126 · calculationBefore Exp sum argument factorial step3 := mul(2, 3) After Exp sum argument factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125 S127 · calculationBefore Exp sum argument factorial step4 := mul(6, 4) After Exp sum argument factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S126 S128 · calculationBefore Exp sum argument factorial step5 := mul(24, 5) After Exp sum argument factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127 S129 · calculationBefore Exp sum argument factorial step6 := mul(120, 6) After Exp sum argument factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S128 S130 · calculationBefore Exp sum argument factorial step7 := mul(720, 7) After Exp sum argument factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S129 S131 · calculationBefore Exp sum argument factorial step8 := mul(5040, 8) After Exp sum argument factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S130 S132 · calculationBefore Exp sum argument coefficient := div(1/21435888100000000, 40320) After Exp sum argument coefficient = 1/864295008192000000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S131 S133 · impliesBefore source roles = [[9/10, -10/11], a_n=1/n! for EVERY n>=0] After 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)] Conditions: none For any fixed real z,choose N+1>2|z|;later absolute term ratios are below1/2 and a geometric tail proves absolute convergence.This is not finite-prefix extrapolation or omission of |z|. Step identifiers and dependenciesRule: Exp_WHOLE_factorial_law_ALL_fixed_real_arguments · Uses: G001, S132 S134 · impliesBefore source roles = [[9/10, -10/11, 8], E(x),E(y) absolutely convergent] After 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)] Conditions: none Multiply the finite degree-n convolution by n!:choosing k of n factors gives the binomial expansion for arbitrary n.Absolute convergence permits regrouping the double sum into the Cauchy product,establishing the whole addition law,not extrapolating one degree. Step identifiers and dependenciesRule: Exp_ALL_degree_binomial_and_absolute_Cauchy_product · Uses: G001, S133 ## Reverse checks - S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S132 · premise_and_result_audit — Recompute Exp sum argument coefficient with exact rational arithmetic, not rounded decimals. - S131 · premise_and_result_audit — Recompute Exp sum argument factorial step8 with exact rational arithmetic, not rounded decimals. - S130 · premise_and_result_audit — Recompute Exp sum argument factorial step7 with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Recompute Exp sum argument factorial step6 with exact rational arithmetic, not rounded decimals. - S128 · premise_and_result_audit — Recompute Exp sum argument factorial step5 with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Recompute Exp sum argument factorial step4 with exact rational arithmetic, not rounded decimals. - S126 · premise_and_result_audit — Recompute Exp sum argument factorial step3 with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Recompute Exp sum argument factorial step2 with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute Exp sum argument factorial step1 with exact rational arithmetic, not rounded decimals. - S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S122 · premise_and_result_audit — Recompute Exp ORIGINAL arguments sum with exact rational arithmetic, not rounded decimals. - S121 · premise_and_result_audit — Recompute Exp complete declared-degree convolution with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute Exp convolution coefficient term8 with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute Exp convolution denominator8 with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S117 · premise_and_result_audit — Recompute Exp convolution left8 factorial step8 with exact rational arithmetic, not rounded decimals. - S116 · premise_and_result_audit — Recompute Exp convolution left8 factorial step7 with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Recompute Exp convolution left8 factorial step6 with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute Exp convolution left8 factorial step5 with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute Exp convolution left8 factorial step4 with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute Exp convolution left8 factorial step3 with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Recompute Exp convolution left8 factorial step2 with exact rational arithmetic, not rounded decimals. - S110 · premise_and_result_audit — Recompute Exp convolution left8 factorial step1 with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute Exp convolution numerator8 with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S106 · premise_and_result_audit — Recompute Exp convolution coefficient term7 with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Recompute Exp convolution denominator7 with exact rational arithmetic, not rounded decimals. - S104 · premise_and_result_audit — Recompute Exp convolution right7 factorial step1 with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute Exp convolution left7 factorial step7 with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Recompute Exp convolution left7 factorial step6 with exact rational arithmetic, not rounded decimals. - S101 · premise_and_result_audit — Recompute Exp convolution left7 factorial step5 with exact rational arithmetic, not rounded decimals. - S100 · premise_and_result_audit — Recompute Exp convolution left7 factorial step4 with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Recompute Exp convolution left7 factorial step3 with exact rational arithmetic, not rounded decimals. - S098 · premise_and_result_audit — Recompute Exp convolution left7 factorial step2 with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute Exp convolution left7 factorial step1 with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute Exp convolution numerator7 with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S093 · premise_and_result_audit — Recompute Exp convolution coefficient term6 with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute Exp convolution denominator6 with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute Exp convolution right6 factorial step2 with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute Exp convolution right6 factorial step1 with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute Exp convolution left6 factorial step6 with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute Exp convolution left6 factorial step5 with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Recompute Exp convolution left6 factorial step4 with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute Exp convolution left6 factorial step3 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute Exp convolution left6 factorial step2 with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute Exp convolution left6 factorial step1 with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute Exp convolution numerator6 with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S080 · premise_and_result_audit — Recompute Exp convolution coefficient term5 with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute Exp convolution denominator5 with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute Exp convolution right5 factorial step3 with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute Exp convolution right5 factorial step2 with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute Exp convolution right5 factorial step1 with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute Exp convolution left5 factorial step5 with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute Exp convolution left5 factorial step4 with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute Exp convolution left5 factorial step3 with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute Exp convolution left5 factorial step2 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute Exp convolution left5 factorial step1 with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute Exp convolution numerator5 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S067 · premise_and_result_audit — Recompute Exp convolution coefficient term4 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute Exp convolution denominator4 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute Exp convolution right4 factorial step4 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute Exp convolution right4 factorial step3 with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute Exp convolution right4 factorial step2 with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute Exp convolution right4 factorial step1 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Exp convolution left4 factorial step4 with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute Exp convolution left4 factorial step3 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute Exp convolution left4 factorial step2 with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute Exp convolution left4 factorial step1 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Exp convolution numerator4 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Recompute Exp convolution coefficient term3 with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Exp convolution denominator3 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute Exp convolution right3 factorial step5 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Exp convolution right3 factorial step4 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Exp convolution right3 factorial step3 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Exp convolution right3 factorial step2 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Exp convolution right3 factorial step1 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Exp convolution left3 factorial step3 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Exp convolution left3 factorial step2 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Exp convolution left3 factorial step1 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Exp convolution numerator3 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Recompute Exp convolution coefficient term2 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Exp convolution denominator2 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Exp convolution right2 factorial step6 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Exp convolution right2 factorial step5 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Exp convolution right2 factorial step4 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Exp convolution right2 factorial step3 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Exp convolution right2 factorial step2 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Exp convolution right2 factorial step1 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Exp convolution left2 factorial step2 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Exp convolution left2 factorial step1 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Exp convolution numerator2 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Recompute Exp convolution coefficient term1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Exp convolution denominator1 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Exp convolution right1 factorial step7 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Exp convolution right1 factorial step6 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Exp convolution right1 factorial step5 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Exp convolution right1 factorial step4 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Exp convolution right1 factorial step3 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Exp convolution right1 factorial step2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Exp convolution right1 factorial step1 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Exp convolution left1 factorial step1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Exp convolution numerator1 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute Exp convolution coefficient term0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Exp convolution denominator0 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 (9/10, -10/11, 8, [[0, 156250/13504609503], [1, -12500/136410197], [2, 1125/3543122], [3, -405/644204], [4, 729/937024], [5, -6561/10648000], [6, 59049/193600000], [7, -531441/6160000000], [8, 4782969/448000000000]], 1/864295008192000000000, 1/864295008192000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 45bc35f2b0f694737e2db283633e59e49be33d07c1383ecf04fcc103b4e6d4d1 - candidate record SHA-256: 2b5f0166f1bb5a57291c1fda0c1be7e762eb84b4f565c37993679cc8f345cc40 - semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386 - question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r02-real-exponential-function-0059.json)