# Question 24213: 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-0063) 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. R0047-24213 · q-20261015-advance-20261113-r02-real-exponential-function-0063 ## 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=13/14, y=-14/15, declared degree=12. 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=13/14, y=-14/15, declared degree=12. 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 (13/14)^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 (-14/15)^12 After Exp convolution y power12 = 56693912375296/129746337890625 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, 56693912375296/129746337890625) After Exp convolution numerator0 = 56693912375296/129746337890625 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 right0 factorial step9 := mul(40320, 9) After Exp convolution right0 factorial step9 = 362880 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 right0 factorial step10 := mul(362880, 10) After Exp convolution right0 factorial step10 = 3628800 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 · calculationBefore Exp convolution right0 factorial step11 := mul(3628800, 11) After Exp convolution right0 factorial step11 = 39916800 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, S015 S017 · calculationBefore Exp convolution right0 factorial step12 := mul(39916800, 12) After Exp convolution right0 factorial step12 = 479001600 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, S016 S018 · calculationBefore Exp convolution denominator0 := mul(1, 479001600) After Exp convolution denominator0 = 479001600 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 coefficient term0 := div(56693912375296/129746337890625, 479001600) After Exp convolution coefficient term0 = 7909306972/8670299029541015625 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 · impliesBefore (13/14)^1 After Exp convolution x power1 = 13/14 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S019 S021 · impliesBefore (-14/15)^11 After Exp convolution y power11 = -4049565169664/8649755859375 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S020 S022 · calculationBefore Exp convolution numerator1 := mul(13/14, -4049565169664/8649755859375) After Exp convolution numerator1 = -3760310514688/8649755859375 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 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, S022 S024 · 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, S023 S025 · 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, S024 S026 · 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, S025 S027 · 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, S026 S028 · 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, S027 S029 · 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, S028 S030 · 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, S029 S031 · calculationBefore Exp convolution right1 factorial step8 := mul(5040, 8) After Exp convolution right1 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, S030 S032 · calculationBefore Exp convolution right1 factorial step9 := mul(40320, 9) After Exp convolution right1 factorial step9 = 362880 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 right1 factorial step10 := mul(362880, 10) After Exp convolution right1 factorial step10 = 3628800 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 right1 factorial step11 := mul(3628800, 11) After Exp convolution right1 factorial step11 = 39916800 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 denominator1 := mul(1, 39916800) After Exp convolution denominator1 = 39916800 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 coefficient term1 := div(-3760310514688/8649755859375, 39916800) After Exp convolution coefficient term1 = -2098387564/192673311767578125 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 · impliesBefore (13/14)^2 After Exp convolution x power2 = 169/196 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S036 S038 · impliesBefore (-14/15)^10 After Exp convolution y power10 = 289254654976/576650390625 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S037 S039 · calculationBefore Exp convolution numerator2 := mul(169/196, 289254654976/576650390625) After Exp convolution numerator2 = 249408350464/576650390625 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 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, S039 S041 · 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, S040 S042 · 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, S041 S043 · 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, S042 S044 · 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, S043 S045 · 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, S044 S046 · 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, S045 S047 · 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, S046 S048 · calculationBefore Exp convolution right2 factorial step7 := mul(720, 7) After Exp convolution right2 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, S047 S049 · calculationBefore Exp convolution right2 factorial step8 := mul(5040, 8) After Exp convolution right2 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, S048 S050 · calculationBefore Exp convolution right2 factorial step9 := mul(40320, 9) After Exp convolution right2 factorial step9 = 362880 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 right2 factorial step10 := mul(362880, 10) After Exp convolution right2 factorial step10 = 3628800 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 denominator2 := mul(2, 3628800) After Exp convolution denominator2 = 7257600 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 coefficient term2 := div(249408350464/576650390625, 7257600) After Exp convolution coefficient term2 = 139178767/2335434082031250 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 · impliesBefore (13/14)^3 After Exp convolution x power3 = 2197/2744 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S053 S055 · impliesBefore (-14/15)^9 After Exp convolution y power9 = -20661046784/38443359375 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 · calculationBefore Exp convolution numerator3 := mul(2197/2744, -20661046784/38443359375) After Exp convolution numerator3 = -16542390592/38443359375 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, S055 S057 · 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, S056 S058 · 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, S057 S059 · 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, S058 S060 · 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, S059 S061 · 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, S060 S062 · 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, S061 S063 · 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, S062 S064 · 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, S063 S065 · calculationBefore Exp convolution right3 factorial step6 := mul(120, 6) After Exp convolution right3 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, S064 S066 · calculationBefore Exp convolution right3 factorial step7 := mul(720, 7) After Exp convolution right3 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, S065 S067 · calculationBefore Exp convolution right3 factorial step8 := mul(5040, 8) After Exp convolution right3 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, S066 S068 · calculationBefore Exp convolution right3 factorial step9 := mul(40320, 9) After Exp convolution right3 factorial step9 = 362880 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, S067 S069 · calculationBefore Exp convolution denominator3 := mul(6, 362880) After Exp convolution denominator3 = 2177280 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, S068 S070 · calculationBefore Exp convolution coefficient term3 := div(-16542390592/38443359375, 2177280) After Exp convolution coefficient term3 = -36924979/186834726562500 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 · impliesBefore (13/14)^4 After Exp convolution x power4 = 28561/38416 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S070 S072 · impliesBefore (-14/15)^8 After Exp convolution y power8 = 1475789056/2562890625 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S071 S073 · calculationBefore Exp convolution numerator4 := mul(28561/38416, 1475789056/2562890625) After Exp convolution numerator4 = 1097199376/2562890625 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 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, S073 S075 · 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, S074 S076 · 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, S075 S077 · 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, S076 S078 · 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, S077 S079 · 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, S078 S080 · 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, S079 S081 · 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, S080 S082 · calculationBefore Exp convolution right4 factorial step5 := mul(24, 5) After Exp convolution right4 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, S081 S083 · calculationBefore Exp convolution right4 factorial step6 := mul(120, 6) After Exp convolution right4 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, S082 S084 · calculationBefore Exp convolution right4 factorial step7 := mul(720, 7) After Exp convolution right4 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, S083 S085 · calculationBefore Exp convolution right4 factorial step8 := mul(5040, 8) After Exp convolution right4 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, S084 S086 · calculationBefore Exp convolution denominator4 := mul(24, 40320) After Exp convolution denominator4 = 967680 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 coefficient term4 := div(1097199376/2562890625, 967680) After Exp convolution coefficient term4 = 9796423/22143375000000 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 · impliesBefore (13/14)^5 After Exp convolution x power5 = 371293/537824 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S087 S089 · impliesBefore (-14/15)^7 After Exp convolution y power7 = -105413504/170859375 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S088 S090 · calculationBefore Exp convolution numerator5 := mul(371293/537824, -105413504/170859375) After Exp convolution numerator5 = -72773428/170859375 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 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, S090 S092 · 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, S091 S093 · 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, S092 S094 · 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, S093 S095 · 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, S094 S096 · 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, S095 S097 · 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, S096 S098 · 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, S097 S099 · calculationBefore Exp convolution right5 factorial step4 := mul(6, 4) After Exp convolution right5 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, S098 S100 · calculationBefore Exp convolution right5 factorial step5 := mul(24, 5) After Exp convolution right5 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, S099 S101 · calculationBefore Exp convolution right5 factorial step6 := mul(120, 6) After Exp convolution right5 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, S100 S102 · calculationBefore Exp convolution right5 factorial step7 := mul(720, 7) After Exp convolution right5 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, S101 S103 · calculationBefore Exp convolution denominator5 := mul(120, 5040) After Exp convolution denominator5 = 604800 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 coefficient term5 := div(-72773428/170859375, 604800) After Exp convolution coefficient term5 = -2599051/3690562500000 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 · impliesBefore (13/14)^6 After Exp convolution x power6 = 4826809/7529536 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S104 S106 · impliesBefore (-14/15)^6 After Exp convolution y power6 = 7529536/11390625 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S105 S107 · calculationBefore Exp convolution numerator6 := mul(4826809/7529536, 7529536/11390625) After Exp convolution numerator6 = 4826809/11390625 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, S106 S108 · 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, S107 S109 · 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, S108 S110 · 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, S109 S111 · 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, S110 S112 · 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, S111 S113 · 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, S112 S114 · 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, S113 S115 · 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, S114 S116 · calculationBefore Exp convolution right6 factorial step3 := mul(2, 3) After Exp convolution right6 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, S115 S117 · calculationBefore Exp convolution right6 factorial step4 := mul(6, 4) After Exp convolution right6 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, S116 S118 · calculationBefore Exp convolution right6 factorial step5 := mul(24, 5) After Exp convolution right6 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, S117 S119 · calculationBefore Exp convolution right6 factorial step6 := mul(120, 6) After Exp convolution right6 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, S118 S120 · calculationBefore Exp convolution denominator6 := mul(720, 720) After Exp convolution denominator6 = 518400 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 convolution coefficient term6 := div(4826809/11390625, 518400) After Exp convolution coefficient term6 = 4826809/5904900000000 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 · impliesBefore (13/14)^7 After Exp convolution x power7 = 62748517/105413504 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S121 S123 · impliesBefore (-14/15)^5 After Exp convolution y power5 = -537824/759375 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 convolution numerator7 := mul(62748517/105413504, -537824/759375) After Exp convolution numerator7 = -62748517/148837500 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 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, S124 S126 · 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, S125 S127 · 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, S126 S128 · 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, S127 S129 · 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, S128 S130 · 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, S129 S131 · 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, S130 S132 · 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, S131 S133 · calculationBefore Exp convolution right7 factorial step2 := mul(1, 2) After Exp convolution right7 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, S132 S134 · calculationBefore Exp convolution right7 factorial step3 := mul(2, 3) After Exp convolution right7 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, S133 S135 · calculationBefore Exp convolution right7 factorial step4 := mul(6, 4) After Exp convolution right7 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, S134 S136 · calculationBefore Exp convolution right7 factorial step5 := mul(24, 5) After Exp convolution right7 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, S135 S137 · calculationBefore Exp convolution denominator7 := mul(5040, 120) After Exp convolution denominator7 = 604800 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, S136 S138 · calculationBefore Exp convolution coefficient term7 := div(-62748517/148837500, 604800) After Exp convolution coefficient term7 = -62748517/90016920000000 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, S137 S139 · impliesBefore (13/14)^8 After Exp convolution x power8 = 815730721/1475789056 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S138 S140 · impliesBefore (-14/15)^4 After Exp convolution y power4 = 38416/50625 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S139 S141 · calculationBefore Exp convolution numerator8 := mul(815730721/1475789056, 38416/50625) After Exp convolution numerator8 = 815730721/1944810000 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, S140 S142 · 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, S141 S143 · 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, S142 S144 · 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, S143 S145 · 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, S144 S146 · 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, S145 S147 · 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, S146 S148 · 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, S147 S149 · 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, S148 S150 · calculationBefore Exp convolution right8 factorial step1 := mul(1, 1) After Exp convolution right8 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, S149 S151 · calculationBefore Exp convolution right8 factorial step2 := mul(1, 2) After Exp convolution right8 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, S150 S152 · calculationBefore Exp convolution right8 factorial step3 := mul(2, 3) After Exp convolution right8 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, S151 S153 · calculationBefore Exp convolution right8 factorial step4 := mul(6, 4) After Exp convolution right8 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, S152 S154 · calculationBefore Exp convolution denominator8 := mul(40320, 24) After Exp convolution denominator8 = 967680 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, S153 S155 · calculationBefore Exp convolution coefficient term8 := div(815730721/1944810000, 967680) After Exp convolution coefficient term8 = 815730721/1881953740800000 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, S154 S156 · impliesBefore (13/14)^9 After Exp convolution x power9 = 10604499373/20661046784 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S155 S157 · impliesBefore (-14/15)^3 After Exp convolution y power3 = -2744/3375 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S156 S158 · calculationBefore Exp convolution numerator9 := mul(10604499373/20661046784, -2744/3375) After Exp convolution numerator9 = -10604499373/25412184000 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, S157 S159 · calculationBefore Exp convolution left9 factorial step1 := mul(1, 1) After Exp convolution left9 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, S158 S160 · calculationBefore Exp convolution left9 factorial step2 := mul(1, 2) After Exp convolution left9 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, S159 S161 · calculationBefore Exp convolution left9 factorial step3 := mul(2, 3) After Exp convolution left9 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, S160 S162 · calculationBefore Exp convolution left9 factorial step4 := mul(6, 4) After Exp convolution left9 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, S161 S163 · calculationBefore Exp convolution left9 factorial step5 := mul(24, 5) After Exp convolution left9 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, S162 S164 · calculationBefore Exp convolution left9 factorial step6 := mul(120, 6) After Exp convolution left9 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, S163 S165 · calculationBefore Exp convolution left9 factorial step7 := mul(720, 7) After Exp convolution left9 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, S164 S166 · calculationBefore Exp convolution left9 factorial step8 := mul(5040, 8) After Exp convolution left9 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, S165 S167 · calculationBefore Exp convolution left9 factorial step9 := mul(40320, 9) After Exp convolution left9 factorial step9 = 362880 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, S166 S168 · calculationBefore Exp convolution right9 factorial step1 := mul(1, 1) After Exp convolution right9 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, S167 S169 · calculationBefore Exp convolution right9 factorial step2 := mul(1, 2) After Exp convolution right9 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, S168 S170 · calculationBefore Exp convolution right9 factorial step3 := mul(2, 3) After Exp convolution right9 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, S169 S171 · calculationBefore Exp convolution denominator9 := mul(362880, 6) After Exp convolution denominator9 = 2177280 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, S170 S172 · calculationBefore Exp convolution coefficient term9 := div(-10604499373/25412184000, 2177280) After Exp convolution coefficient term9 = -10604499373/55329439979520000 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, S171 S173 · impliesBefore (13/14)^10 After Exp convolution x power10 = 137858491849/289254654976 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S172 S174 · impliesBefore (-14/15)^2 After Exp convolution y power2 = 196/225 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S173 S175 · calculationBefore Exp convolution numerator10 := mul(137858491849/289254654976, 196/225) After Exp convolution numerator10 = 137858491849/332052537600 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, S174 S176 · calculationBefore Exp convolution left10 factorial step1 := mul(1, 1) After Exp convolution left10 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, S175 S177 · calculationBefore Exp convolution left10 factorial step2 := mul(1, 2) After Exp convolution left10 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, S176 S178 · calculationBefore Exp convolution left10 factorial step3 := mul(2, 3) After Exp convolution left10 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, S177 S179 · calculationBefore Exp convolution left10 factorial step4 := mul(6, 4) After Exp convolution left10 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, S178 S180 · calculationBefore Exp convolution left10 factorial step5 := mul(24, 5) After Exp convolution left10 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, S179 S181 · calculationBefore Exp convolution left10 factorial step6 := mul(120, 6) After Exp convolution left10 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, S180 S182 · calculationBefore Exp convolution left10 factorial step7 := mul(720, 7) After Exp convolution left10 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, S181 S183 · calculationBefore Exp convolution left10 factorial step8 := mul(5040, 8) After Exp convolution left10 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, S182 S184 · calculationBefore Exp convolution left10 factorial step9 := mul(40320, 9) After Exp convolution left10 factorial step9 = 362880 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, S183 S185 · calculationBefore Exp convolution left10 factorial step10 := mul(362880, 10) After Exp convolution left10 factorial step10 = 3628800 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, S184 S186 · calculationBefore Exp convolution right10 factorial step1 := mul(1, 1) After Exp convolution right10 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, S185 S187 · calculationBefore Exp convolution right10 factorial step2 := mul(1, 2) After Exp convolution right10 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, S186 S188 · calculationBefore Exp convolution denominator10 := mul(3628800, 2) After Exp convolution denominator10 = 7257600 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, S187 S189 · calculationBefore Exp convolution coefficient term10 := div(137858491849/332052537600, 7257600) After Exp convolution coefficient term10 = 137858491849/2409904496885760000 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, S188 S190 · impliesBefore (13/14)^11 After Exp convolution x power11 = 1792160394037/4049565169664 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S189 S191 · impliesBefore (-14/15)^1 After Exp convolution y power1 = -14/15 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S190 S192 · calculationBefore Exp convolution numerator11 := mul(1792160394037/4049565169664, -14/15) After Exp convolution numerator11 = -1792160394037/4338819824640 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, S191 S193 · calculationBefore Exp convolution left11 factorial step1 := mul(1, 1) After Exp convolution left11 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, S192 S194 · calculationBefore Exp convolution left11 factorial step2 := mul(1, 2) After Exp convolution left11 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, S193 S195 · calculationBefore Exp convolution left11 factorial step3 := mul(2, 3) After Exp convolution left11 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, S194 S196 · calculationBefore Exp convolution left11 factorial step4 := mul(6, 4) After Exp convolution left11 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, S195 S197 · calculationBefore Exp convolution left11 factorial step5 := mul(24, 5) After Exp convolution left11 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, S196 S198 · calculationBefore Exp convolution left11 factorial step6 := mul(120, 6) After Exp convolution left11 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, S197 S199 · calculationBefore Exp convolution left11 factorial step7 := mul(720, 7) After Exp convolution left11 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, S198 S200 · calculationBefore Exp convolution left11 factorial step8 := mul(5040, 8) After Exp convolution left11 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, S199 S201 · calculationBefore Exp convolution left11 factorial step9 := mul(40320, 9) After Exp convolution left11 factorial step9 = 362880 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, S200 S202 · calculationBefore Exp convolution left11 factorial step10 := mul(362880, 10) After Exp convolution left11 factorial step10 = 3628800 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, S201 S203 · calculationBefore Exp convolution left11 factorial step11 := mul(3628800, 11) After Exp convolution left11 factorial step11 = 39916800 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, S202 S204 · calculationBefore Exp convolution right11 factorial step1 := mul(1, 1) After Exp convolution right11 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, S203 S205 · calculationBefore Exp convolution denominator11 := mul(39916800, 1) After Exp convolution denominator11 = 39916800 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, S204 S206 · calculationBefore Exp convolution coefficient term11 := div(-1792160394037/4338819824640, 39916800) After Exp convolution coefficient term11 = -1792160394037/173191803176189952000 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, S205 S207 · impliesBefore (13/14)^12 After Exp convolution x power12 = 23298085122481/56693912375296 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S206 S208 · impliesBefore (-14/15)^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, S207 S209 · calculationBefore Exp convolution numerator12 := mul(23298085122481/56693912375296, 1) After Exp convolution numerator12 = 23298085122481/56693912375296 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, S208 S210 · calculationBefore Exp convolution left12 factorial step1 := mul(1, 1) After Exp convolution left12 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, S209 S211 · calculationBefore Exp convolution left12 factorial step2 := mul(1, 2) After Exp convolution left12 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, S210 S212 · calculationBefore Exp convolution left12 factorial step3 := mul(2, 3) After Exp convolution left12 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, S211 S213 · calculationBefore Exp convolution left12 factorial step4 := mul(6, 4) After Exp convolution left12 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, S212 S214 · calculationBefore Exp convolution left12 factorial step5 := mul(24, 5) After Exp convolution left12 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, S213 S215 · calculationBefore Exp convolution left12 factorial step6 := mul(120, 6) After Exp convolution left12 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, S214 S216 · calculationBefore Exp convolution left12 factorial step7 := mul(720, 7) After Exp convolution left12 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, S215 S217 · calculationBefore Exp convolution left12 factorial step8 := mul(5040, 8) After Exp convolution left12 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, S216 S218 · calculationBefore Exp convolution left12 factorial step9 := mul(40320, 9) After Exp convolution left12 factorial step9 = 362880 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, S217 S219 · calculationBefore Exp convolution left12 factorial step10 := mul(362880, 10) After Exp convolution left12 factorial step10 = 3628800 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, S218 S220 · calculationBefore Exp convolution left12 factorial step11 := mul(3628800, 11) After Exp convolution left12 factorial step11 = 39916800 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, S219 S221 · calculationBefore Exp convolution left12 factorial step12 := mul(39916800, 12) After Exp convolution left12 factorial step12 = 479001600 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, S220 S222 · 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, S221 S223 · calculationBefore Exp convolution denominator12 := mul(479001600, 1) After Exp convolution denominator12 = 479001600 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, S222 S224 · calculationBefore Exp convolution coefficient term12 := div(23298085122481/56693912375296, 479001600) After Exp convolution coefficient term12 = 23298085122481/27156474738026584473600 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, S223 S225 · calculationBefore Exp complete declared-degree convolution := sum(7909306972/8670299029541015625, -2098387564/192673311767578125, 139178767/2335434082031250, -36924979/186834726562500, 9796423/22143375000000, -2599051/3690562500000, 4826809/5904900000000, -62748517/90016920000000, 815730721/1881953740800000, -10604499373/55329439979520000, 137858491849/2409904496885760000, -1792160394037/173191803176189952000, 23298085122481/27156474738026584473600) After Exp complete declared-degree convolution = 1/3523453147278219257625600000000000000 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, S224 S226 · calculationBefore Exp ORIGINAL arguments sum := sum(13/14, -14/15) After Exp ORIGINAL arguments sum = -1/210 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, S225 S227 · impliesBefore (-1/210)^12 After Exp sum argument declared power = 1/7355827511386641000000000000 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S226 S228 · 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, S227 S229 · 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, S228 S230 · 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, S229 S231 · 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, S230 S232 · 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, S231 S233 · 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, S232 S234 · 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, S233 S235 · 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, S234 S236 · calculationBefore Exp sum argument factorial step9 := mul(40320, 9) After Exp sum argument factorial step9 = 362880 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, S235 S237 · calculationBefore Exp sum argument factorial step10 := mul(362880, 10) After Exp sum argument factorial step10 = 3628800 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, S236 S238 · calculationBefore Exp sum argument factorial step11 := mul(3628800, 11) After Exp sum argument factorial step11 = 39916800 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, S237 S239 · calculationBefore Exp sum argument factorial step12 := mul(39916800, 12) After Exp sum argument factorial step12 = 479001600 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, S238 S240 · calculationBefore Exp sum argument coefficient := div(1/7355827511386641000000000000, 479001600) After Exp sum argument coefficient = 1/3523453147278219257625600000000000000 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, S239 S241 · impliesBefore source roles = [[13/14, -14/15], 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, S240 S242 · impliesBefore source roles = [[13/14, -14/15, 12], 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, S241 ## Reverse checks - S242 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S241 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S240 · premise_and_result_audit — Recompute Exp sum argument coefficient with exact rational arithmetic, not rounded decimals. - S239 · premise_and_result_audit — Recompute Exp sum argument factorial step12 with exact rational arithmetic, not rounded decimals. - S238 · premise_and_result_audit — Recompute Exp sum argument factorial step11 with exact rational arithmetic, not rounded decimals. - S237 · premise_and_result_audit — Recompute Exp sum argument factorial step10 with exact rational arithmetic, not rounded decimals. - S236 · premise_and_result_audit — Recompute Exp sum argument factorial step9 with exact rational arithmetic, not rounded decimals. - S235 · premise_and_result_audit — Recompute Exp sum argument factorial step8 with exact rational arithmetic, not rounded decimals. - S234 · premise_and_result_audit — Recompute Exp sum argument factorial step7 with exact rational arithmetic, not rounded decimals. - S233 · premise_and_result_audit — Recompute Exp sum argument factorial step6 with exact rational arithmetic, not rounded decimals. - S232 · premise_and_result_audit — Recompute Exp sum argument factorial step5 with exact rational arithmetic, not rounded decimals. - S231 · premise_and_result_audit — Recompute Exp sum argument factorial step4 with exact rational arithmetic, not rounded decimals. - S230 · premise_and_result_audit — Recompute Exp sum argument factorial step3 with exact rational arithmetic, not rounded decimals. - S229 · premise_and_result_audit — Recompute Exp sum argument factorial step2 with exact rational arithmetic, not rounded decimals. - S228 · premise_and_result_audit — Recompute Exp sum argument factorial step1 with exact rational arithmetic, not rounded decimals. - S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S226 · premise_and_result_audit — Recompute Exp ORIGINAL arguments sum with exact rational arithmetic, not rounded decimals. - S225 · premise_and_result_audit — Recompute Exp complete declared-degree convolution with exact rational arithmetic, not rounded decimals. - S224 · premise_and_result_audit — Recompute Exp convolution coefficient term12 with exact rational arithmetic, not rounded decimals. - S223 · premise_and_result_audit — Recompute Exp convolution denominator12 with exact rational arithmetic, not rounded decimals. - S222 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S221 · premise_and_result_audit — Recompute Exp convolution left12 factorial step12 with exact rational arithmetic, not rounded decimals. - S220 · premise_and_result_audit — Recompute Exp convolution left12 factorial step11 with exact rational arithmetic, not rounded decimals. - S219 · premise_and_result_audit — Recompute Exp convolution left12 factorial step10 with exact rational arithmetic, not rounded decimals. - S218 · premise_and_result_audit — Recompute Exp convolution left12 factorial step9 with exact rational arithmetic, not rounded decimals. - S217 · premise_and_result_audit — Recompute Exp convolution left12 factorial step8 with exact rational arithmetic, not rounded decimals. - S216 · premise_and_result_audit — Recompute Exp convolution left12 factorial step7 with exact rational arithmetic, not rounded decimals. - S215 · premise_and_result_audit — Recompute Exp convolution left12 factorial step6 with exact rational arithmetic, not rounded decimals. - S214 · premise_and_result_audit — Recompute Exp convolution left12 factorial step5 with exact rational arithmetic, not rounded decimals. - S213 · premise_and_result_audit — Recompute Exp convolution left12 factorial step4 with exact rational arithmetic, not rounded decimals. - S212 · premise_and_result_audit — Recompute Exp convolution left12 factorial step3 with exact rational arithmetic, not rounded decimals. - S211 · premise_and_result_audit — Recompute Exp convolution left12 factorial step2 with exact rational arithmetic, not rounded decimals. - S210 · premise_and_result_audit — Recompute Exp convolution left12 factorial step1 with exact rational arithmetic, not rounded decimals. - S209 · premise_and_result_audit — Recompute Exp convolution numerator12 with exact rational arithmetic, not rounded decimals. - S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S206 · premise_and_result_audit — Recompute Exp convolution coefficient term11 with exact rational arithmetic, not rounded decimals. - S205 · premise_and_result_audit — Recompute Exp convolution denominator11 with exact rational arithmetic, not rounded decimals. - S204 · premise_and_result_audit — Recompute Exp convolution right11 factorial step1 with exact rational arithmetic, not rounded decimals. - S203 · premise_and_result_audit — Recompute Exp convolution left11 factorial step11 with exact rational arithmetic, not rounded decimals. - S202 · premise_and_result_audit — Recompute Exp convolution left11 factorial step10 with exact rational arithmetic, not rounded decimals. - S201 · premise_and_result_audit — Recompute Exp convolution left11 factorial step9 with exact rational arithmetic, not rounded decimals. - S200 · premise_and_result_audit — Recompute Exp convolution left11 factorial step8 with exact rational arithmetic, not rounded decimals. - S199 · premise_and_result_audit — Recompute Exp convolution left11 factorial step7 with exact rational arithmetic, not rounded decimals. - S198 · premise_and_result_audit — Recompute Exp convolution left11 factorial step6 with exact rational arithmetic, not rounded decimals. - S197 · premise_and_result_audit — Recompute Exp convolution left11 factorial step5 with exact rational arithmetic, not rounded decimals. - S196 · premise_and_result_audit — Recompute Exp convolution left11 factorial step4 with exact rational arithmetic, not rounded decimals. - S195 · premise_and_result_audit — Recompute Exp convolution left11 factorial step3 with exact rational arithmetic, not rounded decimals. - S194 · premise_and_result_audit — Recompute Exp convolution left11 factorial step2 with exact rational arithmetic, not rounded decimals. - S193 · premise_and_result_audit — Recompute Exp convolution left11 factorial step1 with exact rational arithmetic, not rounded decimals. - S192 · premise_and_result_audit — Recompute Exp convolution numerator11 with exact rational arithmetic, not rounded decimals. - S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S189 · premise_and_result_audit — Recompute Exp convolution coefficient term10 with exact rational arithmetic, not rounded decimals. - S188 · premise_and_result_audit — Recompute Exp convolution denominator10 with exact rational arithmetic, not rounded decimals. - S187 · premise_and_result_audit — Recompute Exp convolution right10 factorial step2 with exact rational arithmetic, not rounded decimals. - S186 · premise_and_result_audit — Recompute Exp convolution right10 factorial step1 with exact rational arithmetic, not rounded decimals. - S185 · premise_and_result_audit — Recompute Exp convolution left10 factorial step10 with exact rational arithmetic, not rounded decimals. - S184 · premise_and_result_audit — Recompute Exp convolution left10 factorial step9 with exact rational arithmetic, not rounded decimals. - S183 · premise_and_result_audit — Recompute Exp convolution left10 factorial step8 with exact rational arithmetic, not rounded decimals. - S182 · premise_and_result_audit — Recompute Exp convolution left10 factorial step7 with exact rational arithmetic, not rounded decimals. - S181 · premise_and_result_audit — Recompute Exp convolution left10 factorial step6 with exact rational arithmetic, not rounded decimals. - S180 · premise_and_result_audit — Recompute Exp convolution left10 factorial step5 with exact rational arithmetic, not rounded decimals. - S179 · premise_and_result_audit — Recompute Exp convolution left10 factorial step4 with exact rational arithmetic, not rounded decimals. - S178 · premise_and_result_audit — Recompute Exp convolution left10 factorial step3 with exact rational arithmetic, not rounded decimals. - S177 · premise_and_result_audit — Recompute Exp convolution left10 factorial step2 with exact rational arithmetic, not rounded decimals. - S176 · premise_and_result_audit — Recompute Exp convolution left10 factorial step1 with exact rational arithmetic, not rounded decimals. - S175 · premise_and_result_audit — Recompute Exp convolution numerator10 with exact rational arithmetic, not rounded decimals. - S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S172 · premise_and_result_audit — Recompute Exp convolution coefficient term9 with exact rational arithmetic, not rounded decimals. - S171 · premise_and_result_audit — Recompute Exp convolution denominator9 with exact rational arithmetic, not rounded decimals. - S170 · premise_and_result_audit — Recompute Exp convolution right9 factorial step3 with exact rational arithmetic, not rounded decimals. - S169 · premise_and_result_audit — Recompute Exp convolution right9 factorial step2 with exact rational arithmetic, not rounded decimals. - S168 · premise_and_result_audit — Recompute Exp convolution right9 factorial step1 with exact rational arithmetic, not rounded decimals. - S167 · premise_and_result_audit — Recompute Exp convolution left9 factorial step9 with exact rational arithmetic, not rounded decimals. - S166 · premise_and_result_audit — Recompute Exp convolution left9 factorial step8 with exact rational arithmetic, not rounded decimals. - S165 · premise_and_result_audit — Recompute Exp convolution left9 factorial step7 with exact rational arithmetic, not rounded decimals. - S164 · premise_and_result_audit — Recompute Exp convolution left9 factorial step6 with exact rational arithmetic, not rounded decimals. - S163 · premise_and_result_audit — Recompute Exp convolution left9 factorial step5 with exact rational arithmetic, not rounded decimals. - S162 · premise_and_result_audit — Recompute Exp convolution left9 factorial step4 with exact rational arithmetic, not rounded decimals. - S161 · premise_and_result_audit — Recompute Exp convolution left9 factorial step3 with exact rational arithmetic, not rounded decimals. - S160 · premise_and_result_audit — Recompute Exp convolution left9 factorial step2 with exact rational arithmetic, not rounded decimals. - S159 · premise_and_result_audit — Recompute Exp convolution left9 factorial step1 with exact rational arithmetic, not rounded decimals. - S158 · premise_and_result_audit — Recompute Exp convolution numerator9 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S155 · premise_and_result_audit — Recompute Exp convolution coefficient term8 with exact rational arithmetic, not rounded decimals. - S154 · premise_and_result_audit — Recompute Exp convolution denominator8 with exact rational arithmetic, not rounded decimals. - S153 · premise_and_result_audit — Recompute Exp convolution right8 factorial step4 with exact rational arithmetic, not rounded decimals. - S152 · premise_and_result_audit — Recompute Exp convolution right8 factorial step3 with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Recompute Exp convolution right8 factorial step2 with exact rational arithmetic, not rounded decimals. - S150 · premise_and_result_audit — Recompute Exp convolution right8 factorial step1 with exact rational arithmetic, not rounded decimals. - S149 · premise_and_result_audit — Recompute Exp convolution left8 factorial step8 with exact rational arithmetic, not rounded decimals. - S148 · premise_and_result_audit — Recompute Exp convolution left8 factorial step7 with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Recompute Exp convolution left8 factorial step6 with exact rational arithmetic, not rounded decimals. - S146 · premise_and_result_audit — Recompute Exp convolution left8 factorial step5 with exact rational arithmetic, not rounded decimals. - S145 · premise_and_result_audit — Recompute Exp convolution left8 factorial step4 with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute Exp convolution left8 factorial step3 with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Recompute Exp convolution left8 factorial step2 with exact rational arithmetic, not rounded decimals. - S142 · premise_and_result_audit — Recompute Exp convolution left8 factorial step1 with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Recompute Exp convolution numerator8 with exact rational arithmetic, not rounded decimals. - S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S138 · premise_and_result_audit — Recompute Exp convolution coefficient term7 with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Recompute Exp convolution denominator7 with exact rational arithmetic, not rounded decimals. - S136 · premise_and_result_audit — Recompute Exp convolution right7 factorial step5 with exact rational arithmetic, not rounded decimals. - S135 · premise_and_result_audit — Recompute Exp convolution right7 factorial step4 with exact rational arithmetic, not rounded decimals. - S134 · premise_and_result_audit — Recompute Exp convolution right7 factorial step3 with exact rational arithmetic, not rounded decimals. - S133 · premise_and_result_audit — Recompute Exp convolution right7 factorial step2 with exact rational arithmetic, not rounded decimals. - S132 · premise_and_result_audit — Recompute Exp convolution right7 factorial step1 with exact rational arithmetic, not rounded decimals. - S131 · premise_and_result_audit — Recompute Exp convolution left7 factorial step7 with exact rational arithmetic, not rounded decimals. - S130 · premise_and_result_audit — Recompute Exp convolution left7 factorial step6 with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Recompute Exp convolution left7 factorial step5 with exact rational arithmetic, not rounded decimals. - S128 · premise_and_result_audit — Recompute Exp convolution left7 factorial step4 with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Recompute Exp convolution left7 factorial step3 with exact rational arithmetic, not rounded decimals. - S126 · premise_and_result_audit — Recompute Exp convolution left7 factorial step2 with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Recompute Exp convolution left7 factorial step1 with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute Exp convolution numerator7 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S121 · premise_and_result_audit — Recompute Exp convolution coefficient term6 with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute Exp convolution denominator6 with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute Exp convolution right6 factorial step6 with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute Exp convolution right6 factorial step5 with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute Exp convolution right6 factorial step4 with exact rational arithmetic, not rounded decimals. - S116 · premise_and_result_audit — Recompute Exp convolution right6 factorial step3 with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Recompute Exp convolution right6 factorial step2 with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute Exp convolution right6 factorial step1 with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute Exp convolution left6 factorial step6 with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute Exp convolution left6 factorial step5 with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Recompute Exp convolution left6 factorial step4 with exact rational arithmetic, not rounded decimals. - S110 · premise_and_result_audit — Recompute Exp convolution left6 factorial step3 with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute Exp convolution left6 factorial step2 with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Recompute Exp convolution left6 factorial step1 with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Recompute Exp convolution numerator6 with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S104 · premise_and_result_audit — Recompute Exp convolution coefficient term5 with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute Exp convolution denominator5 with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Recompute Exp convolution right5 factorial step7 with exact rational arithmetic, not rounded decimals. - S101 · premise_and_result_audit — Recompute Exp convolution right5 factorial step6 with exact rational arithmetic, not rounded decimals. - S100 · premise_and_result_audit — Recompute Exp convolution right5 factorial step5 with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Recompute Exp convolution right5 factorial step4 with exact rational arithmetic, not rounded decimals. - S098 · premise_and_result_audit — Recompute Exp convolution right5 factorial step3 with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute Exp convolution right5 factorial step2 with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute Exp convolution right5 factorial step1 with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Recompute Exp convolution left5 factorial step5 with exact rational arithmetic, not rounded decimals. - S094 · premise_and_result_audit — Recompute Exp convolution left5 factorial step4 with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Recompute Exp convolution left5 factorial step3 with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute Exp convolution left5 factorial step2 with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute Exp convolution left5 factorial step1 with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute Exp convolution numerator5 with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute Exp convolution coefficient term4 with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute Exp convolution denominator4 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute Exp convolution right4 factorial step8 with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute Exp convolution right4 factorial step7 with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute Exp convolution right4 factorial step6 with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute Exp convolution right4 factorial step5 with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute Exp convolution right4 factorial step4 with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute Exp convolution right4 factorial step3 with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute Exp convolution right4 factorial step2 with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute Exp convolution right4 factorial step1 with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute Exp convolution left4 factorial step4 with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute Exp convolution left4 factorial step3 with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute Exp convolution left4 factorial step2 with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute Exp convolution left4 factorial step1 with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute Exp convolution numerator4 with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S070 · premise_and_result_audit — Recompute Exp convolution coefficient term3 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Exp convolution denominator3 with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute Exp convolution right3 factorial step9 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute Exp convolution right3 factorial step8 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute Exp convolution right3 factorial step7 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute Exp convolution right3 factorial step6 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute Exp convolution right3 factorial step5 with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute Exp convolution right3 factorial step4 with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute Exp convolution right3 factorial step3 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Exp convolution right3 factorial step2 with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute Exp convolution right3 factorial step1 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute Exp convolution left3 factorial step3 with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute Exp convolution left3 factorial step2 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Exp convolution left3 factorial step1 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Exp convolution numerator3 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Recompute Exp convolution coefficient term2 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute Exp convolution denominator2 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Exp convolution right2 factorial step10 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Exp convolution right2 factorial step9 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Exp convolution right2 factorial step8 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Exp convolution right2 factorial step7 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Exp convolution right2 factorial step6 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Exp convolution right2 factorial step5 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Exp convolution right2 factorial step4 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Exp convolution right2 factorial step3 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Exp convolution right2 factorial step2 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Exp convolution right2 factorial step1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Exp convolution left2 factorial step2 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Exp convolution left2 factorial step1 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Exp convolution numerator2 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute Exp convolution coefficient term1 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Exp convolution denominator1 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Exp convolution right1 factorial step11 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Exp convolution right1 factorial step10 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Exp convolution right1 factorial step9 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Exp convolution right1 factorial step8 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Exp convolution right1 factorial step7 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Exp convolution right1 factorial step6 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Exp convolution right1 factorial step5 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Exp convolution right1 factorial step4 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Exp convolution right1 factorial step3 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Exp convolution right1 factorial step2 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Exp convolution right1 factorial step1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Exp convolution left1 factorial step1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Exp convolution numerator1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute Exp convolution coefficient term0 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Exp convolution denominator0 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Exp convolution right0 factorial step12 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Exp convolution right0 factorial step11 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Exp convolution right0 factorial step10 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 (13/14, -14/15, 12, [[0, 7909306972/8670299029541015625], [1, -2098387564/192673311767578125], [2, 139178767/2335434082031250], [3, -36924979/186834726562500], [4, 9796423/22143375000000], [5, -2599051/3690562500000], [6, 4826809/5904900000000], [7, -62748517/90016920000000], [8, 815730721/1881953740800000], [9, -10604499373/55329439979520000], [10, 137858491849/2409904496885760000], [11, -1792160394037/173191803176189952000], [12, 23298085122481/27156474738026584473600]], 1/3523453147278219257625600000000000000, 1/3523453147278219257625600000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement) 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: 30f9c1c5f3d0e3f15d0cce7e8931b2389f6273df1ded0a6b0b518bc8700bd5c0 - candidate record SHA-256: 0438ea42ce8fbc417eb6881978179bf4fd1e6b21b90fd34d3ba93100307b617b - 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-0063.json)