# Question 24225: 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-0075) 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. R0047-24225 · q-20261015-advance-20261113-r02-real-exponential-function-0075 ## 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=25/26, y=-26/27, declared degree=3. 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=25/26, y=-26/27, declared degree=3. 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 (25/26)^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 (-26/27)^3 After Exp convolution y power3 = -17576/19683 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, -17576/19683) After Exp convolution numerator0 = -17576/19683 Conditions: 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 denominator0 := mul(1, 6) After Exp convolution denominator0 = 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, S008 S010 · calculationBefore Exp convolution coefficient term0 := div(-17576/19683, 6) After Exp convolution coefficient term0 = -8788/59049 Conditions: 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 · impliesBefore (25/26)^1 After Exp convolution x power1 = 25/26 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S010 S012 · impliesBefore (-26/27)^2 After Exp convolution y power2 = 676/729 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S011 S013 · calculationBefore Exp convolution numerator1 := mul(25/26, 676/729) After Exp convolution numerator1 = 650/729 Conditions: 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 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, S013 S015 · 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, S014 S016 · 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, S015 S017 · calculationBefore Exp convolution denominator1 := mul(1, 2) After Exp convolution denominator1 = 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, S016 S018 · calculationBefore Exp convolution coefficient term1 := div(650/729, 2) After Exp convolution coefficient term1 = 325/729 Conditions: 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 · impliesBefore (25/26)^2 After Exp convolution x power2 = 625/676 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S018 S020 · impliesBefore (-26/27)^1 After Exp convolution y power1 = -26/27 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 · calculationBefore Exp convolution numerator2 := mul(625/676, -26/27) After Exp convolution numerator2 = -625/702 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Exp convolution 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, S021 S023 · 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, S022 S024 · 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, S023 S025 · calculationBefore Exp convolution denominator2 := mul(2, 1) After Exp convolution denominator2 = 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 coefficient term2 := div(-625/702, 2) After Exp convolution coefficient term2 = -625/1404 Conditions: 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 · impliesBefore (25/26)^3 After Exp convolution x power3 = 15625/17576 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S026 S028 · impliesBefore (-26/27)^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, S027 S029 · calculationBefore Exp convolution numerator3 := mul(15625/17576, 1) After Exp convolution numerator3 = 15625/17576 Conditions: 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 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, S029 S031 · 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, S030 S032 · 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, S031 S033 · 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, S032 S034 · calculationBefore Exp convolution denominator3 := mul(6, 1) After Exp convolution denominator3 = 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, S033 S035 · calculationBefore Exp convolution coefficient term3 := div(15625/17576, 6) After Exp convolution coefficient term3 = 15625/105456 Conditions: 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 complete declared-degree convolution := sum(-8788/59049, 325/729, -625/1404, 15625/105456) After Exp complete declared-degree convolution = -1/2075690448 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Exp ORIGINAL arguments sum := sum(25/26, -26/27) After Exp ORIGINAL arguments sum = -1/702 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore (-1/702)^3 After Exp sum argument declared power = -1/345948408 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 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, S038 S040 · 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, S039 S041 · 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, S040 S042 · calculationBefore Exp sum argument coefficient := div(-1/345948408, 6) After Exp sum argument coefficient = -1/2075690448 Conditions: 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 · impliesBefore source roles = [[25/26, -26/27], 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, S042 S044 · impliesBefore source roles = [[25/26, -26/27, 3], 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, S043 ## Reverse checks - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Recompute Exp sum argument coefficient with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Exp sum argument factorial step3 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Exp sum argument factorial step2 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Exp sum argument factorial step1 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 — Recompute Exp ORIGINAL arguments sum with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Exp complete declared-degree convolution with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Exp convolution coefficient term3 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Exp convolution denominator3 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute Exp convolution left3 factorial step3 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Exp convolution left3 factorial step2 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Exp convolution left3 factorial step1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Exp convolution numerator3 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute Exp convolution coefficient term2 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Exp convolution denominator2 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Exp convolution right2 factorial step1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Exp convolution left2 factorial step2 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Exp convolution left2 factorial step1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Exp convolution numerator2 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute Exp convolution coefficient term1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Exp convolution denominator1 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Exp convolution right1 factorial step2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Exp convolution right1 factorial step1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Exp convolution left1 factorial step1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Exp convolution numerator1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute Exp convolution coefficient term0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Exp convolution denominator0 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 (25/26, -26/27, 3, [[0, -8788/59049], [1, 325/729], [2, -625/1404], [3, 15625/105456]], -1/2075690448, -1/2075690448, 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: bb50d60cc8a5d3fbf057b5a8aad201c4838814edbb8de40c4c1fe6d0f57a8b1c - candidate record SHA-256: c53c1a9be585945b225552b65f5e18685fb842a1c54f4c7959529a2278d3d091 - 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-0075.json)