Each coefficient field $X_j$ of the modified logarithm is a holomorphic symplectic field with a global holomorphic primitive $H_j$ on the star-shaped domain, and $X_1=X_{\tilde{A}+\tilde{B}}$

Given: The analytic splitting data; the fields $X_j$ with components $(X_j{)}_k$; the bounds ${\max }_i\mid (X_j{)}_i\mid \le {\beta }_j$ on $D_j$ (Bounds for the truncated logarithm: $\mid L_N(h)z_i\mathrm{/}h\mid \le 8M\mathrm{/}3$ and $\mid X_j\mid \le (8M\mathrm{/}3)(C_{0}Mj\mathrm{/}\rho {)}^{j-1}$); the matrix $J$ and the form ${\omega }_{\mathbb{C}}$ (Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket, A phase domain, its canonical one-form and its standard symplectic two-form).

Coordinate bracket matrix formula: for holomorphic $f,g$, $\{f,g\}=(\mathrm{\nabla }f{)}^{\mathsf{T}}J\mathrm{\nabla }g$, and the bracket is $\mathbb{C}$-bilinear and antisymmetric with $\{Z_k,Z_l\}=J_{kl}$ constant (Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits clause 2, Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket, Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes).

Derivation of the bracket: for the coefficient operators $D_j$ of the logarithm, $D_j\{f,g\}=\{D_{j}f,g\}+\{f,D_{j}g\}$ for holomorphic $f,g$, and $X_j$ acts on $f$ as $X_{j}f=D_{j}f$ (The binomial family of the splitting pullback is multiplicative and bracket-preserving, and its logarithm is a derivation clause 3, The truncated binomial family $T_h^{s,N}$, the truncated logarithm $L_N(h)$, its coefficient fields $X_j$ and the truncations $X^{[N]}$).

Realness of the extensions and their flows. By the distance-domain estimate in Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket, $D(\rho )$ is star-shaped about $0$. Fix $\xi \in D(\rho )\cap {\mathbb{R}}^{2d}$; the set of real $t$ for which $t\xi \in D(\rho )$ has a connected open interval containing $[0,1]$. The function $g_A(t):=\mathrm{Im}\tilde{A}(t\xi )$ is real analytic on that interval, since it is the restriction of a holomorphic function to a complex line. For $t$ in a neighborhood of $0$, $t\xi \in U$ because $0\in U$ and $U$ is open, so $g_A(t)=0$ as $\tilde{A}{\mathrm{\mid }}_U=A$ and $A$ is real-valued. To justify continuation, let $S$ be the set of points where $g_A$ vanishes on a neighborhood. It is nonempty and open; if $t_j\in S$ converges to an interior point $t$, every derivative $g_A^{(k)}(t_j)$ is zero, so continuity of all derivatives gives $g_A^{(k)}(t)=0$ for every $k$, and the real-analytic Taylor expansion at $t$ makes $g_A$ zero near $t$. Thus $S$ is also closed in the interval and equals the whole interval. This proves $\tilde{A}(\xi )\in \mathbb{R}$; the identical argument applies to $\tilde{B}$. The restrictions of these extensions to the open real domain $D(\rho )\cap {\mathbb{R}}^{2d}$ are real-valued, so their real-coordinate partial derivatives are real; holomorphic differentiability identifies those derivatives with the complex coordinate derivatives. Hence $X_{\tilde{A}}=J\mathrm{\nabla }\tilde{A}$ and $X_{\tilde{B}}=J\mathrm{\nabla }\tilde{B}$ are real on the real locus.

For either field $F=X_{\tilde{A}}$ or $X_{\tilde{B}}$, the Picard iterates in The flow of a bounded holomorphic field on a polydisc: existence in complex time, displacement bound, holomorphy, and the composition law starting from real $Z$ remain real when $t$ is real: the initial iterate is real, and each next iterate is $Z+t{\int }_0^{1}F({\gamma }_m(\sigma t))d\sigma$, an integral of real vectors by induction. Their limit is real because ${\mathbb{R}}^{2d}$ is closed in ${\mathbb{C}}^{2d}$. Thus each local real-time flow preserves real points; uniqueness and the local gluing in that lemma preserve this property along every real continuation where the flow is defined. The three factors defining ${\mathrm{\Psi }}_h$ therefore preserve real points for real $h$, as do all defined iterates ${\mathrm{\Psi }}_h^m$.

Closed one-forms on star-shaped real domains have potentials, with the explicit radial potential $\varphi (x)={\int }_0^1⟨F(tx),x⟩dt$ for centre $0$ (Poincare's lemma on a star-shaped domain: every closed C1 field is exact, Star-shaped open subsets of Euclidean space); the complex domain $D_j\subseteq {\mathbb{C}}^{2d}\cong {\mathbb{R}}^{4d}$ is open and star-shaped about $0$ (Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket), and the real and imaginary parts of a closed complex one-form with $C^1$ coefficients are closed real one-forms.

Holomorphy criterion: a complex-valued $C^1$ function $H$ on an open subset of ${\mathbb{C}}^{2d}$ viewed as a real domain with ${\mathrm{\partial }}_{{\bar{Z}}_l}H=0$ for all $l$ is holomorphic, since then $dH$ is $\mathbb{C}$-linear; and the zero-normalized radial primitive $H(Z)={\int }_0^1\alpha (tZ)(Z)dt$ is real-valued on the real part of its star-shaped domain whenever $\alpha$ is real on real points and real tangent vectors, because its integrand is then real (Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket, Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits clause 2, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder).