Optimal finite truncation gives an exponential local splitting defect

Uniform field and flow radius. Fix $N$ and let $u$ be a complex parameter with $\mathrm{\mid }u\mathrm{\mid }\le R_N=r_N\mathrm{/}{C}_1$. By [F1], summing a geometric series gives $\mathrm{\parallel }{X}^{[N]}(u)\mathrm{\parallel }\le \sqrt{m}(8M\mathrm{/}3)\mathrm{/}(1-C_1^{-1})\le 4\sqrt{m}M=:B$ on $D(\rho \mathrm{/}4)$. At each $Z\in D(\rho \mathrm{/}8)$ the polydisc of radius $s=\rho \mathrm{/}(16\sqrt{m})$ lies in $D(\rho \mathrm{/}4)$. The flow radius $s\mathrm{/}(8mB)=\rho \mathrm{/}(512m^{2}M)$ is at least $R_N$ since $N\ge 2$ and $C_1=1024m^2$. Picard iteration is also holomorphic in $u$ (each iterate is a holomorphic parameter integral and the contraction is uniform); hence the time-$u$ map of the field $X^{[N]}(u)$ is holomorphic in $u$ for $\mathrm{\mid }u\mathrm{\mid }<R_N$. Its displacement is at most $B\mathrm{\mid }u\mathrm{\mid }$. The splitting displacement is at most $2M\mathrm{\mid }u\mathrm{\mid }$. All estimates hold uniformly on the initial-data domain.

Finite coefficient matching. In degree at most $N$, $u{X}^{[N]}(u)$ agrees with $L(u)$ as operators by [F2]. Terms in the exponential of order above $N$ cannot change these degrees because both operators start in degree one. The time-$u$ pullback of a fixed field $X^{[N]}(u)$ has Taylor coefficients ${\sum }_{k\ge 0}{u}^k(X^{[N]}(u){)}^k\mathrm{/}k!$: differentiating $f({\phi }^t(Z))$ repeatedly in $t$ gives $X^{k}f$ and the holomorphic Taylor expansion supplies the coefficients. Therefore the two analytic coordinate maps agree through degree $N$. This argument uses finite jets and makes no assertion that the infinite formal logarithm converges.

Cauchy tail. For each coordinate the difference $E_N(u,Z)$ of the two maps divided by $u$ extends holomorphically at zero, has its first $N$ coefficients zero, and is bounded by $2M+B$ on $\mathrm{\mid }u\mathrm{\mid }<R_N$. Cauchy estimates on smaller circles, followed by taking their radii to $R_N$, give $\mathrm{\mid }{E}_N(h,Z)\mathrm{\mid }\le (2M+B)(h\mathrm{/}{R}_N{)}^N\mathrm{/}(1-h\mathrm{/}{R}_N)$ for $0<h<R_N$. The actual map difference is $h{E}_N(h,Z)$; multiplying its coordinate bound by $\sqrt{m}$ gives the Euclidean bound.

Optimal integer choice. For $N=\lfloor a\mathrm{/}h\rfloor$, $h\mathrm{/}{R}_N=C_1{C}_{0}MNh\mathrm{/}\rho \le C_1{C}_0\mathrm{/}{C}_2=1\mathrm{/}(16e)<1\mathrm{/}2$ by [F4]. Also $N\ge a\mathrm{/}(2h)$ when $h\le a\mathrm{/}3$. The logarithm definition and the exponential addition law give $2^{-N}=\exp (-N\log 2)$; therefore $N\log 2\ge c\mathrm{/}h$ implies $2^{-N}\le \exp (-c\mathrm{/}h)$ by monotonicity of $\exp$. Thus $(h\mathrm{/}{R}_N{)}^N\le 2^{-N}\le e^{-c\mathrm{/}h}$. The denominator in step 3.1 is at least $1\mathrm{/}2$, yielding the asserted defect. Interchanging $A,B$ leaves all bounds and formal arguments unchanged.