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R0047-24280 · q-20261015-advance-20261113-r02-real-exponential-function-0080
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 real argument x=-8. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, 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 real argument x=-8. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).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.
Before
source roles = [exp-zero-inverse-positivity]
After
derived = [[x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, 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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Exp ORIGINAL real argument half := div(-8, 2)
After
Exp ORIGINAL real argument half = -4
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [[-8], 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|.
Rule: Exp_WHOLE_factorial_law_ALL_fixed_real_arguments · Uses: G001, S002
Before
source roles = [[-8, -4], 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.
Rule: Exp_ALL_degree_binomial_and_absolute_Cauchy_product · Uses: G001, S003
Before
source roles = [-8, -4]
After
derived = [E(0)=1, E(x)E(-x)=1, E(x/2)E(-x/2)=1, E(x/2)!=0, E(x)=E(x/2)^2>0]
Conditions: none
At0,the constant term1 and all later zero terms give E(0)=1.Addition gives inverse products1,hence a nonzero real half-value.Only then does its square give strict positivity.No rational exact value at nonzero input,derivative,log or ODE is claimed.
Rule: Exp_zero_identity_then_inverse_half_NONZERO_then_square · Uses: G001, S004
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Recompute Exp ORIGINAL real argument half with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(-8, E(0)=1, E(-8)*E(8)=1, E(-4)*E(4)=1, E(-4)!=0, E(-8)=E(-4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: a6e0afaec1daeb18cc5310e87b265848ba141f33debdd215be243d8ec812e73bcandidate record SHA-256: 7ae9792f52dea00fff0f6872e3fe8135fbbc32e33213707a33814a37cf1bc4d8semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED