For k=n+8, the sqrt/log quotient error is <=2*2^-k and exponential truncation error <=2^-k; hence |u_n-exp(L/sqrt2)|<=7*2^-k<2^-n and 1<=u_n<=2.
Method: native-order. Induction: none. Risk: critical.
This is a human-readable planning contract, not a parsed kernel formula or accepted proof.