Proof. An ambient homeomorphism of the plane takes the attached-disc pair to \(D=[-2,0]\times[-1,1]\) and \(D'=[-2,3]\times[-1,1]\). Indeed, compatible parametrizations of the attaching arc and the remaining boundary arcs extend over both discs by Jordan–Schoenflies; the resulting outer boundary parametrization extends across the exterior. In this model choose \[0<c<c'<a'<a<b<b'<d'<d<3,\qquad 0<\delta<\epsilon.\] Use the following horizontal intervals: \[\begin{array}{c|c|c} &\text{compact piece}&\text{open piece}\\ \hline B_0,\ U_0&[-2-\epsilon,a]&(-2-\delta,a')\\ A,\ U_A&[c,d]&(c',d')\\ B_1,\ U_1&[b,3+\epsilon]&(b',3+\delta). \end{array}\] All compact rectangles have vertical interval \([-1-\epsilon,1+\epsilon]\), and all open rectangles have interval \((-1-\delta,1+\delta)\). Take \(a,\epsilon\) small enough that \(B_0\) is in the prescribed neighborhood. The closed fringes are disjoint compact sets, as is seen from their horizontal intervals \([a',b']\) and \([-2-\delta,c']\cup[d',3+\delta]\). Pull the sets back by the homeomorphism. Compact containment and positive separation persist. The compact unions in the statement are unions of two disjoint filled Jordan discs, hence have connected complement and are polynomially convex. ◻

Proof. Choose a smooth function \(\chi\) on the plane, zero near \(\overline{U_A\setminus U_B}\) and one near \(\overline{U_B\setminus U_A}\), with bounded first derivative. On \(U_A\) extend \(\chi u\) by zero off \(C\); on \(U_B\) extend \((\chi-1)u\) similarly. Both extensions have planar \(\bar\partial\) derivative \(u\bar\partial\chi\) on \(C\) and zero elsewhere in \(\Omega=U_A\cup U_B\). Extend this bounded coefficient by zero to \(\mathbb C\) and set \[v(s,t)=\frac1\pi\int_C \frac{u(\zeta,t)\,\partial_{\bar\zeta}\chi(\zeta)} {s-\zeta}\,dA(\zeta).\] The distributional identity \(\partial_{\bar s}(1/(\pi s))=\delta_0\) gives the required \(\bar\partial\) equation. If \(d\) is the diameter of \(\Omega\), then \[\|v\|_{\Omega\times P} \le 2d\|\bar\partial\chi\|_\infty\|u\|_{C\times P}.\] The kernel is locally integrable, so \(v\) is continuous in \(s\) and holomorphic in \(t\). Thus \(R_Au=\chi u-v\) and \(R_Bu=(\chi-1)u-v\) are holomorphic, have the asserted bounds, and differ by \(u\). No boundary continuity of \(u\) is needed. ◻

Proof of Proposition 4. First consider complete strips. Choose \(\rho>0\) so small that \(5\rho<b\), that \(\tau=\sigma_2^{-1}\circ\sigma_1\) is defined near \(\overline{\Delta}_{3\rho}^2\) using the inverse branch at the origin, and that \(D\tau\) is invertible there. Exact axis agreement gives \(\tau(z,0)=(z,0)\). The matrices \(T_0(z)=D\tau(z,0)\) and their inverses are bounded on \(|z|\le2\rho\). Choose a biholomorphic \(\sigma_2\) chart \(Q_c\) about \((0,c)\) whose first coordinate has modulus less than \(\rho/16\), and take a relatively compact smaller image \(V\) as the witness neighborhood.

Fix input data as in Definition 3. Choose an open polydisc \(P_*\) with \(P_0\Subset P_*\Subset\operatorname{int}P\). In the chart \(Q_c\) write \(g=\sigma_2^{-1}\circ f=(g_1,g_2)\). There is a planar neighborhood \(N\) of \(D\) where \(g\) is defined for parameters in a neighborhood of \(\overline P_*\) and \(|g_1|<\rho/8\). No smallness of \(g_2-c\) beyond membership in \(Q_c\) is used in the overlap calculation.

Choose the pieces of Lemma 5. Products of polynomially convex compact sets are polynomially convex. Scalar Oka–Weil approximation therefore gives a polynomial \(h_1\) in the planar and parameter variables approximating \(g_1\) on \(B_0\times\overline P_*\) and zero on \(B_1\times\overline P_*\). Its error \(\epsilon_1\) can be arbitrarily small; in particular, \(|h_1|<\rho/2\) on their union. Independently, for every sufficiently small \(\eta>0\) there is a polynomial \(h_2\) satisfying \[\|h_2-g_2\|_{D\times\overline P_*}<\eta,\qquad \|h_2\|_{A\times\overline P_*}<\eta.\] Let \(h=(h_1,h_2)\) and \(C=U_A\cap U_B\). On \(U_B\times P_*\), \(T=T_0(h_1)\) is holomorphic and boundedly invertible. We seek corrections satisfying \[\tau(h+\alpha)=h+\beta,\qquad \alpha=R_Au,\quad\beta=T R_Bu.\] By Lemma 6, this is the fixed-point equation \[u=\mathcal T(u):= R_Au+T^{-1}\bigl(h-\tau(h+R_Au)\bigr)\] in \(H^\infty(C\times P_*,\mathbb C^2)\).

On the overlap \(|h_1|<\rho/2\) and \(|h_2|<\eta\). Bounded first and second derivatives of \(\tau\), and its exact axis identity, give a constant \(M\ge1\) with \[\|T\|,\|T^{-1}\|\le M,\quad \|\tau(h)-h\|\le M\eta,\quad \|D\tau(h+\alpha)-T\|\le M(\eta+\|\alpha\|).\] These constants do not depend on the degrees or uncontrolled coordinates of the polynomials. If \(\kappa\ge1\) bounds both splitting operators and \(L_0=2M^2+1\), then on \(\|u\|\le L_0\eta\), \[\|\mathcal T(0)\|\le M^2\eta,\qquad \|D\mathcal T(u)\|\le M^2\kappa(1+\kappa L_0)\eta.\] For small \(\eta\) the last bound is at most \(1/2\), the ball maps strictly into itself, and all transition evaluations remain in \(\Delta_{3\rho}^2\). The contraction theorem gives \[\|\alpha\|\le\kappa L_0\eta,\qquad \|\beta\|\le M\kappa L_0\eta.\]

On \(U_A\times P_*\) only the second coordinate of \(h+\alpha\) has to be small, and on \(U_B\times P_*\) only the first coordinate of \(h+\beta\) has to be small. Consequently \(\sigma_1(h+\alpha)\) and \(\sigma_2(h+\beta)\) are defined on these full domains and agree on their overlap. On \(D\times P_*\), the second formula is evaluated close to \(g\) in \(Q_c\) and approximates \(f\) as \(\epsilon_1,\eta\to0\). Since the union of the planar domains contains \(D'\) and \(P_0\Subset P_*\), this proves \(L\).

For the finite-length version choose \(V\), \(g\), and the two polynomials using the limiting chart at \((0,c)\). Convergence on slightly larger compact coordinate neighborhoods gives uniform first and second derivative bounds, uniform inverse-chart domains, and thus uniform constants in the contraction argument. Exact axis agreement gives the same \(O(\eta)\) initial error for every pair. The polynomials are bounded on the fixed bounded planar pieces and \(\overline P_*\), so only a finite range of either long coordinate is used. Choose a sufficiently late pair to contain these ranges and make its chart near \((0,c)\) sufficiently close to the limiting chart. The construction then approximates the original data with the requested accuracy. ◻

Proof. The graph of \(f\) is a closed Stein submanifold of \(S\times X\) and has a Stein neighborhood by the Stein neighborhood theorem (Siu 1976). On that neighborhood consider the vertical tangent bundle for projection to \(S\). A holomorphic frame along the graph extends to holomorphic vertical vector fields by Cartan’s Theorem B. Compose their small-time flows. The resulting map has the prescribed value and a frame as its vertical differential at \(v=0\). Compactness and the holomorphic inverse function theorem give a common radius on a neighborhood of \(K\) and injectivity in each parameter fiber. For the last assertion, a planar domain has the homotopy type of a one-dimensional CW complex, so every complex vector bundle on its product with a polydisc is topologically trivial. The Oka–Grauert principle makes it holomorphically trivial. The same argument applies to a contractible Stein source (Forstnerič 2017). ◻

Proof. Work in local coordinates on \(B\) at \(b_0\). The graph of \(f\) has a Stein neighborhood in \(S\) times the submersion locus. On this neighborhood the surjection from tangent vectors vertical over \(S\) to the pulled-back base tangent bundle admits holomorphic lifts of the base coordinate vector fields: its kernel is coherent, and Cartan’s Theorem B gives surjectivity on global sections. Flow the lifts successively for the small coordinate displacements of \(b\). On a relatively compact source set a common time radius is available. The flows are vertical over \(S\), and their projections to \(B\) are the desired coordinate translations. Their composition is the required family of maps. ◻

Proof. Write the sprays as \(F_A\) and \(F_B\). Compact chart margins give the transition \(\gamma(s,t,v)=F_A(s,t,\cdot)^{-1}(F_B(s,t,v)) =v+E(s,t,v)\) on the overlap, with both \(\|E\|\) and \(\|D_vE\|\) small on a smaller auxiliary polydisc. The derivative estimate follows from the original closeness by Cauchy estimates on the larger polydisc. Equality of the resulting maps asks for \(a=\gamma(b)\). Use \(a=R_Au\), \(b=R_Bu\). Then \[u=E(s,t,R_Bu)\] is a contraction on a small ball in the overlap \(H^\infty\) space. Lemma 6 supplies its uniform bounds and preserves holomorphic dependence on the passive parameters. The resulting corrections remain in the auxiliary polydisc and are as small as the original mismatch. Uniform continuity on the prescribed compact sets gives the asserted approximation. ◻

Proof. We give a finite construction with explicit approximation margins. Choose a compact intermediate parameter polydisc strictly between \(P_0\) and \(P\). Compactness gives a partition of the outer circle into finitely many closed arcs, with endpoint margins, such that all values over each slightly enlarged arc and the intermediate parameter set lie in one coordinate witness neighborhood \(V_j\). At the common endpoint of consecutive arcs the compact set of values lies in \(V_j\cap V_{j+1}\) with a positive target margin. Take a slightly larger outer radius than \(R\) during the construction.

Attach radial spokes. At each endpoint choose a small filled polar rectangle crossing the old outer circle, on which \(f\) and a chart spray from Lemma 8 have their values in both adjacent target charts. The rectangles can be chosen mutually disjoint. Attach to each rectangle a radial segment reaching the larger outer radius. In one of its target charts, continue the coordinate spray constantly along the segment from its attachment point on the outer edge of the rectangle. This defines continuous coordinate functions on the filled rectangle and attached segment, holomorphic on its interior and holomorphic in the passive and auxiliary parameters.

The planar compact set has connected complement. Mergelyan approximation, with parameters, approximates these coordinate functions by polynomials. Here the parameter assertion follows directly by taking finite Taylor sums on a slightly larger parameter polydisc and applying scalar Mergelyan to their coefficients. Consequently the approximation can be made uniform with an auxiliary-parameter margin, and Cauchy estimates preserve the required spray derivatives. On the radial segment its values remain close to the endpoint values for all parameters. A thin neighborhood of each segment therefore also has its values in both adjacent \(V_j\)’s.

Glue this extension to the old chart spray across a radial band inside each original rectangle, using Lemma 10. More explicitly, cut the union domain at a radius slightly larger than \(r\), still inside the old map domain. At the cut it consists only of the disjoint rectangles. A radial cutoff with a fixed transition width gives separated fringes and a bounded splitting constant. The tubes around the longer portions of the spokes may then be narrowed without changing that constant. Gluing errors can be as small as desired. This produces a map near \(K\) and all spokes, close to \(f\) on \(K\), with all spoke values still in the two adjacent target charts.

Fill the sectors. List the finitely many closed polar sectors between successive spokes. At a given step let \(K_{\rm old}\) be the union of the original compact set, all spokes, and the sectors already filled. The next sector \(S\) meets \(K_{\rm old}\) exactly on its bottom arc and its two radial sides. Along these three sides all map values, with a parameter margin, belong to the chosen \(V_j\). This is true initially by the spoke construction, and it remains true after each sufficiently accurate gluing, because the sides of every unfilled sector already belong to \(K_{\rm old}\).

Choose a thin closed U-shaped band \(D\) about these three sides, inside the current map domain, and a slightly enlarged filled sector \(D'\) containing \(S\) in its interior, so that \(D\subset D'\) is a special pair. To see this elementary geometry, straighten the sector to a rectangle: \(D\) is a thick band around its bottom and two side edges, and the missing central rectangle attaches along the U-shaped portion of its boundary. The bands and protrusions can be made as thin as needed. All values near \(D\) lie in \(V_j\). Apply \(L\) to the restriction of a chart spray over the current map, regarding its small auxiliary variables as additional passive parameters. After any prescribed small parameter loss this gives a spray on \(D'\), arbitrarily close to the old spray on \(D\).

We spell out the overlap used to glue. Fix an open band \(W\Subset\operatorname{int}D\) containing \(K_{\rm old}\cap S\). Using a slightly smaller band first, the compact remainders on opposite sides are positively separated. A smooth cutoff can therefore be zero near \(K_{\rm old}\setminus W\) and one near \(S\setminus W\). On a sufficiently thin neighborhood of \(K_{\rm old}\cup S\), its sublevel and superlevel sets give two open domains with separated fringes and overlap contained in \(W\). They may be chosen so that any point of \(K_{\rm old}\) in the new side also lies in the approximation band. The old spray is defined on the first domain, the approximating spray on the second, and they are close on their overlap with an auxiliary margin. Lemma 10 produces the required new map, close to the old map on all of \(K_{\rm old}\).

There are only finitely many spokes and sectors. Fix in advance a finite nested sequence of parameter polydiscs between the intermediate one and \(P_0\). At each step choose errors smaller than the remaining compact target-chart margins and allocate a summable finite part of the requested final error. These choices preserve every unfilled sector’s joining sides. No chart condition on the outer boundary of a newly filled sector is required. At the end we have a map on a neighborhood of the desired enlarged compact set times \(P_0\), with the stated approximation. ◻

Proof. Slightly tilt the line in two different directions through \(x\). Openness and compactness of the original boundary imply that the two tilted discs still have boundary in the good locus, and their tangent directions at \(x\) are transverse. Thicken each disc in a complementary coordinate by a sufficiently small fixed parameter disc. At every boundary point the values for all parameters then lie in one witness neighborhood; finitely many boundary patches give one common positive transverse radius.

For each integer \(n\), apply Proposition 11 once to each thickened disc, enlarging its long coordinate to radius \(n\), retaining a fixed smaller transverse radius, and making its error on the initial product less than \(1/n\). The resulting finite strips converge near their centers to the original coordinate charts. Their limiting long directions are transverse. Remark 7 and the finite-length case of Proposition 4 imply \(L\) at \(x\).

If a proper analytic subset \(E\) contains the exceptional points in a coordinate neighborhood of \(x\), choose a complex line through \(x\) not contained in \(E\). Its intersection with \(E\) is discrete near \(x\), so a sufficiently small circle on that line avoids \(E\). The first assertion applies. ◻

Proof. Use the additive splitting to find \(a,b\) with \(b-a=c\). Shrink the base and fiber domains by \(h\). The splitting bounds and Cauchy estimates in \(v\) imply, with a fixed integer \(p\) enlarged as necessary, \[\|a\|+\|b\|\le Ch^{-p}e,\qquad e=\|c\|,\] including their needed fiber derivatives on intermediate smaller balls. For \(Ch^{-p}e\) sufficiently small relative to these margins, \(\operatorname{id}+b\) has a fiber inverse on a smaller ball, uniformly in the base. The new transition is \[(\operatorname{id}+b)^{-1}\circ\gamma\circ(\operatorname{id}+a) =\operatorname{id}+d.\] The equation determining \(d\) is \[d= c(v+a)-c(v)-\{b(v+d)-b(v)\}.\] The first difference is quadratic in \(e\), up to powers of \(h^{-1}\). The operation in braces is contractive on a small supremum-norm ball, by the first fiber derivative bound. It follows that \[ \|d\|\le Ch^{-p}e^2. \tag{1}\] These estimates involve a fixed finite number of derivatives. Thus one common exponent \(p\) works throughout the iteration; no derivatives of increasing order are requested.

Choose losses \(h_i=h_0 2^{-i}\) whose sum fits within the prescribed margins, leaving intermediate buffers at each step. With \(A=Ch_0^{-p}\), Equation (1) reads \(e_{i+1}\le A2^{pi}e_i^2\). Put \(E_i=A2^{p(i+1)}e_i\). Then \(E_{i+1}\le E_i^2\). Choosing \(E_0<1/2\) gives doubly exponential decay. In particular all sums \(\sum_i h_i^{-q}e_i\) converge for every fixed \(q\), and are as small as desired with \(e_0\).

The successive fiber reparametrizations are composed on the nested domains. Their displacements fit within the reserved buffers and their fiber derivatives have summable deviations from the identity. The compositions therefore converge uniformly. Their limiting transitions are the identity. Composing the two original sprays with these limiting side maps gives equal maps on the overlap. The target approximation assertion follows from the stated continuity control and the smallness of the limiting fiber corrections. ◻

Proof. Put \(r_*=r_1\). Choose \(S>\max(2,r_*)\), and then \(d_0\geq2\) large enough that \[(M/r_*)^d\geq4M\qquad(M\geq S,\ d\geq d_0).\] For example, it suffices to impose this at \(M=S,d=d_0\), since \((M/r_*)^d/M\) increases in both variables on this range. The cone \[{\cal E}=\{u:\max_{j\geq2}|u_j|>\max(S,|u_1|)\}\] is forward invariant under \(H_d\). Indeed, if \(M=\max_{j\geq2}|u_j|\) and \(|u_k|=M\), then \[|H_d(u)_k|\geq(M/r_*)^d-aM\geq3M, \qquad |H_d(u)_1|\leq aM.\] The transverse maximum therefore grows by at least a factor \(3\). In particular, no forward orbit that ends in \(\overline\Delta_1^{\,m}\) can enter \({\cal E}\).

Choose \(C\) larger than \[1,\quad 4a,\quad 16a^2r_*^2,\quad 4S.\] If \(A=\max_{j\geq2}|P_j(x_j)|>C\max(1,|x_1|)\), then \[\max_{j\geq2}|x_j|\leq r_* A^{1/d}\leq r_*\sqrt A.\] For an index \(k\) attaining \(A\), both possible terms \(a|x_{k-1}|\), namely \(a|x_1|\) when \(k=2\) and a transverse coordinate otherwise, are at most \(A/4\). Hence \[|H_d(x)_k|\geq3A/4,\qquad |H_d(x)_1|\leq a r_*\sqrt A\leq A/4.\] Thus \(H_d(x)\in{\cal E}\), a contradiction. This proves (2); the same argument applies at every nonfinal point of the orbit. At its final point the bound is immediate.

Write \(y=H_d(x)\). Applying the power bound at \(y\), or using \(\|y\|_\infty\leq1\) if \(y\) is the final point, gives \[ |y_j|\leq r_j\bigl(C\max(1,a|x_m|)\bigr)^{1/d} \qquad(j\geq2). \tag{5}\] Let \(L_j=\log^+|x_j|\), \(M=L_m\), and \(A_1=\log(r_*\sqrt C+a)\). From \(P_j(x_j)=y_j+a x_{j-1}\) and (5), \[|P_j(x_j)| \leq (r_*\sqrt C+a)\exp(L_{j-1}+M/d).\] Whether or not \(|x_j|\) exceeds \(1\), this implies \[ L_j\leq \log r_j+A_1/d+L_{j-1}/d+M/d^2. \tag{6}\] Put \(A_2=\log r_*+A_1/2\). Iterating (6) to \(j=m\) yields \[M\leq 2A_2+d^{-(m-1)}L_1+ M\sum_{i=0}^{m-2}d^{-2-i} \leq 2A_2+d^{-(m-1)}L_1+\tfrac12M.\] For \(|x_1|\geq r_1\), define \[t=(|x_1|/r_1)^{1/D}\geq1,\qquad A_3=4A_2+2\log r_1.\] It follows that \[M\leq A_3+2\log t,\qquad |y_j|\leq r_j\exp((\log C+A_3)/d)t^{2/d}.\]

Choose a sufficiently small \(\delta>0\) so that \[q:=\max_{2\leq j\leq m} \frac{r_j+\delta}{a(r_{j-1}-\delta)}<1, \qquad r_j-\delta>0.\] Increase \(d_0\), if necessary, so that for all \(d\geq d_0\) \[r_j\exp((\log C+A_3)/d)\leq r_j+\delta,\qquad r_j\bigl((1-q)a(r_{j-1}-\delta)\bigr)^{1/d} \geq r_j-\delta .\] We prove successively that \[|x_j|\geq(r_j-\delta)t^{d^{m-j}}\quad(1\leq j\leq m).\] The assertion for \(j=1\) follows from the definition of \(t\). If it holds for \(j-1\), then \[\frac{|y_j|}{a|x_{j-1}|} \leq \frac{r_j+\delta}{a(r_{j-1}-\delta)} t^{\,2/d-d^{m-j+1}}\leq q.\] Therefore \[|P_j(x_j)|\geq(1-q)a|x_{j-1}|,\] and taking the \(d\)-th root proves the next assertion. This also proves (3). Since \(P_j/(a x_{j-1})=1+y_j/(a x_{j-1})\), the indicated logarithm is well defined; its power series gives the bound \(|\xi_j|\leq\sum_{\nu\geq1}q^\nu/\nu=B\).

There is no root choice in the definition of \(s\). Define it by \[s=\frac{x_m}{\rho_m} \exp\left(-\sum_{k=2}^m\frac{\xi_k}{d^{m-k+1}}\right).\] The equations \[x_j^d=r_j^d a x_{j-1}\exp(\xi_j)\] then prove (4) backwards from \(j=m\) to \(j=1\), using the definition of the positive real numbers \(\rho_j\). ◻

Proof. First prove the finite-radius assertion. When \(m=1\), the one-dimensional boundary circle is covered by finitely many of the neighborhoods in \(L\), and Proposition 11 applies with no passive variables.

Suppose \(m\geq2\). Let \(f\) be the initial map and let \(\varepsilon>0\). Choose \(b>1\) with \(f\) holomorphic near \(\overline\Delta_b^{\,m}\). Choose the radii \(r_j\), the number \(a\), and \(\ell>0\) so that Lemma 16 applies and \[1<r_m<\cdots<r_1<r_1+3\ell<b.\] The derivative of \(H_d\) at zero is the \(d\)-independent map \[L(x)=(a x_m,-a x_1,\ldots,-a x_{m-1}), \qquad \|Lx\|_\infty=a\|x\|_\infty.\] The map \(H_d\) is an automorphism: from \(y=H_d(x)\), first recover \(x_m=y_1/a\), and then successively recover \[x_{j-1}=(P_j(x_j)-y_j)/a,\qquad j=m,m-1,\ldots,2.\] Fix \(R_+>R\), and fix \(N\geq1\) so that \(a^N R_+<1\). Set \[h_d=H_d^{-N}L^N,\qquad \Omega=\Delta_{R_+}^{\,m}.\] Thus \(h_d(\overline\Omega)\subset E_{d,N}\).

On some fixed neighborhood of \(\overline\Delta_1^{\,m}\), \[ h_d\longrightarrow\operatorname{id}\quad(d\longrightarrow\infty). \tag{7}\] Here is a quantitative justification. Choose \(1<b_0<b_1<r_m\), put \(\eta=(b_1-b_0)/3\), and put \(\theta=b_1/r_m<1\). If \(\|L^{-1}y\|_\infty\leq b_0+\eta\) and \(\theta^d/a<\eta\), the inverse recursion above gives \[\|H_d^{-1}(y)-L^{-1}(y)\|_\infty\leq\theta^d/a.\] Indeed, its first recovered coordinate is exact, and each next one differs from the corresponding coordinate of \(L^{-1}y\) by \(P_j(x_j)/a\). Inductively all recovered coordinates have modulus at most \(b_0+2\eta<b_1\), so that \(|P_j(x_j)|\leq\theta^d\). Let \(C_N=\sum_{i=1}^N a^{-i}\), and take \(d\) large enough that \(C_N\theta^d/a<\eta\). Successive comparison of \(H_d^{-r}L^N(z)\) and \(L^{N-r}(z)\), for \(z\in\overline\Delta_{b_0}^{\,m}\), gives the error recursion \[e_0=0,\qquad e_{r+1}\leq a^{-1}e_r+\theta^d/a,\qquad e_r\leq\theta^d\sum_{i=1}^r a^{-i}.\] The imposed bound guarantees the hypothesis of the inverse estimate at every step. At \(r=N\) it proves \(\|h_d-\operatorname{id}\|_{\overline\Delta_{b_0}^{\,m}} \leq C_N\theta^d\), and hence (7).

Choose a number \(b'<b\) larger than \(r_1+2\ell\). Lemma 8 gives a chart spray \[f^\sharp(x,v),\qquad f^\sharp(x,0)=f(x),\] on a neighborhood of \(\overline\Delta_{b'}^{\,m}\times\overline\Delta_\beta^{\,2}\) for some \(\beta>0\). Its vertical differential is invertible and its vertical maps are injective after reducing \(\beta\). The value \(b'\) and this spray are fixed before \(d\) is increased.

Write \(\Phi_d(s,\xi)\) for the right side of (4), now allowing \(\xi\) to vary independently. Use, for example, the three parameter polydiscs \[\Xi_i=\{\xi\in\mathbb C^{m-1}:|\xi_j|\leq B+i\},\qquad i=1,2,3.\] Choose fixed numbers \[r_1<c_0<c_1<c_2<c_3<r_1+\ell.\] On \[c_0\leq |s|^D\leq c_3,\qquad \xi\in\Xi_3,\] one has, uniformly as \(d\to\infty\), \[ |\Phi_{d,j}(s,\xi)|\leq r_j\exp(A_4/d)\quad(j\geq2), \qquad \sup_{\xi,\widetilde\xi\in\Xi_3} \|\Phi_d(s,\xi)-\Phi_d(s,\widetilde\xi)\|_\infty \leq A_5/d, \tag{8}\] where the following fixed constants suffice: \[\begin{aligned} R_0&=2\log r_1+|\log a|,\\ A_4&=|\log a|+R_0+\log c_3+2(B+3),\\ A_5&=4(B+3)r_1e^{A_4/2}. \end{aligned}\] Indeed the recurrence for \(\rho_j\) gives \[|\log\rho_j|\leq R_0,\qquad |\log\rho_j-\log r_j| \leq (|\log a|+R_0)/d .\] The power of \(s\) contributes at most \(\log c_3/d^{j-1}\leq\log c_3/d\). The remaining exponential satisfies \[\sum_{k=2}^j|\xi_k|/d^{j-k+1}\leq2(B+3)/d .\] Moreover \[\partial_{\xi_k}\Phi_{d,j} =d^{-(j-k+1)}\Phi_{d,j}\quad(k\leq j),\qquad \partial_{\xi_k}\Phi_{d,j}=0\quad(k>j),\] Integrating along line segments in the convex polydisc \(\Xi_3\) and using \(\sum_{\nu\geq1}d^{-\nu}\leq2/d\) proves the diameter bound with the displayed \(A_5\). In particular \(\Phi_d\) takes the indicated product into \(\Delta_{b'}^{\,m}\) for large \(d\).

The compact set \(f(\overline\Delta_{b'}^{\,m})\) has a finite cover by neighborhoods on which \(L\) is available. A Lebesgue number for a slightly smaller such cover, uniform continuity of \(f\), and (8) show the following. After increasing \(d\) and decreasing the fixed spray radius, all the values \[f^\sharp(\Phi_d(s,\xi),v),\qquad \xi\in\Xi_3,\quad v\in\overline\Delta_{\beta}^{\,2},\] belong to one of these neighborhoods for each fixed \(s\) on the outer circle \(|s|^D=c_3\). The chosen neighborhood may depend on \(s\). The same increase of \(d\) ensures that \[\sup_{\overline\Delta_1^{\,m}}d_Y(f(h_d(z)),f(z)) <\varepsilon/2,\qquad |h_d(z)_1|<c_1\quad(z\in\overline\Delta_1^{\,m}).\] Require also \(r_j(Cc_2)^{1/d}<b'\) for \(j\geq2\); this is possible since \(r_j<b'\). Fix this value of \(d\), henceforth writing \(h=h_d\).

Apply Proposition 11 to the annulus \[A_s=\{c_0\leq |s|^D\leq c_3\}\] and to the map \(f^\sharp(\Phi_d(s,\xi),v)\), with \(\xi,v\) as passive variables. It gives arbitrarily close approximation on \(A_s\times\Xi_2\times\overline\Delta_{\beta/2}^{\,2}\) by a holomorphic map \(G(s,\xi,v)\) defined out to any prescribed finite outer radius in \(s\), with a smaller fixed positive parameter margin. Choose that radius larger than \(c_3^{1/D}\) and larger than \[\sup\{|s(h(z))|:z\in\overline\Omega,\ |h(z)_1|\geq c_1\}.\] If the set is empty the latter requirement is vacuous. Otherwise this supremum is finite by compactness and the explicit formula for \(s\); \(|s(h(z))|^D=|h(z)_1|\). The inner radius of the domain of \(G\) can be kept strictly below \(c_1^{1/D}\).

On the two open subsets of \(\Omega\) \[A=\{z:|h(z)_1|<c_2\},\qquad B'=\{z:|h(z)_1|>c_1\},\] consider the maps with the common auxiliary variable \(v\) \[F_A(z,v)=f^\sharp(h(z),v),\qquad F_{B'}(z,v)=G(s(h(z)),\xi(h(z)),v).\] The first is defined on all of \(A\): by (2), its transverse coordinates satisfy \[|h(z)_j|\leq r_j(Cc_2)^{1/d}<b'\] after the choices already made. The second is defined on all of \(B'\) by Lemma 16 and the choices of the parameter and outer-radius margins. On their overlap, \(\Phi_d(s(h(z)),\xi(h(z)))=h(z)\); therefore \(F_{B'}\) approximates \(F_A\) arbitrarily closely, uniformly for \(v\) in a fixed smaller polydisc.

Choose a smooth cutoff of \(|h(z)_1|^2\) whose transition is compactly contained in \(c_1<|h(z)_1|<c_2\). On the bounded domain \(\Omega\), all its derivatives needed in Lemma 15 are bounded. These constants, \(d,N\), and all the domain margins have been fixed before choosing the accuracy in Proposition 11. Lemma 15 therefore glues the two maps, by arbitrarily small vertical corrections, on a neighborhood of \(\overline\Delta_R^{\,m}\). On \(\overline\Delta_1^{\,m}\subset A\) the resulting central map differs from \(f\circ h\) by less than \(\varepsilon/2\). Together with (7), this proves the finite-radius assertion.

To obtain an entire map, use this assertion, after affine rescaling, successively on exhausting polydiscs. Choose positive errors with sum less than the prescribed final error, and ensure that the error at each stage is controlled on the previous polydisc. Completeness makes the maps uniformly Cauchy on every compact set. Their limit is holomorphic: near each point, choose a coordinate neighborhood of the limit value; uniform convergence puts all sufficiently late maps in that chart on a smaller source neighborhood, where the usual theorem for uniformly convergent holomorphic functions applies. The same argument gives the prescribed approximation on the initial polydisc. ◻

Proof. We give the gluing detail to avoid assuming a holomorphic extension of the original smooth family. By Lemma 8, a finite cover of \(I\) by short intervals admits fixed graph charts \(E_i(q,v)\) over the corresponding maps. First choose a smooth family of small chart sprays \(H_s^\sharp(q,v)\) centered at \(H_s(q)\), holomorphic in \((q,v)\), on a common small fiber ball. This can be done explicitly along the interval. In one fixed chart, translate its fiber coordinates to the center representing \(H_s\). When changing charts at a division point \(s_0\), express the preceding spray in the new chart and take its fiber displacement at \(s_0\). On a short overlap interpolate this displacement to the fixed displacement at \(s_0\), leaving the moving center unchanged. The two displacements are as close in the first fiber derivative as desired when the overlap is short. The derivatives therefore remain invertible. Finitely many such operations and a smaller common fiber ball give the asserted smooth spray family.

In each division interval write this family in one fixed graph chart. Approximate its coordinate functions by polynomials in \(s\) with coefficients holomorphic in \((q,v)\), retaining any prescribed finite number of derivatives, and interpolate its endpoint jets through order \(M\). This is ordinary approximation in a Banach space of holomorphic functions on a smaller fixed neighborhood: approximate a sufficiently high real derivative uniformly, integrate, and make a finite Hermite correction at the endpoints. The correction tends to zero with the initial approximation error. The target sprays obtained on adjacent intervals consequently have identical \(s\)-jets through order \(M\) at the division point.

Now, after the finitely many polynomials have been fixed, choose a small complex thickness \(\delta>0\) around the real intervals. Extend each interval slightly past its endpoints, so adjacent rectangles overlap within distance \(O(\delta)\) of their common division point. Their spray transition differs from the identity by \(O(\delta^{M+1})\) there, uniformly on a fixed smaller fiber ball and parameter compactum. This is Taylor’s formula applied to the holomorphic coordinate transitions; the constants are fixed before \(\delta\) is chosen.

Additive splitting on these finitely many thin planar rectangles costs at most \(C\delta^{-1}\). Indeed, use cutoffs across the vertical overlap bands, whose first derivatives are \(O(\delta^{-1})\), and the Cauchy–Green operator on a bounded planar set. For simultaneous splitting, each cut contributes a pair of corrections on the entire portion to its left and right; that contribution has zero jump at every other cut. The number of cuts is fixed. The source variables \((q,v)\) are passive for this operator, so no parameter radius is lost.

For clarity, write the consecutive transitions as \(\gamma_i=\operatorname{id}+c_i\), with \(\|c_i\|+\|D_vc_i\|\le A\delta^{M+1}\) on a fixed smaller fiber ball. Let \(R\) be the simultaneous right inverse, of norm at most \(K/\delta\), for the jump operator \((a_i)\mapsto(a_{i+1}-a_i)\). The central maps are identified by the fixed-point equation \[(a_i)=R\bigl((c_i(\,\cdot\,,a_i))_i\bigr).\] It preserves the ball of radius \(2KA\delta^M\) and is contractive there when \(KA\delta^M<1/2\). The corrections are therefore \(O(\delta^M)\). Substituting them into the corresponding sprays gives central maps agreeing on overlaps. Cauchy estimates on rectangles with margins comparable to \(\delta\) bound their correction derivatives through order \(j\) by \(O(\delta^{M-j})\). Derivatives in the other variables use their fixed margins. Take \(M>j+2\) and then \(\delta\) small. Together with the initial polynomial accuracy this gives the prescribed \(C^j\) estimate. ◻

Proof. First shrink any unused margins so that the given map \(H\) is defined on \(V\times U\), where \(V\subset\mathbb C\) is a simply connected bounded neighborhood of \(I\), and \(U\) contains a slightly thickened compact product \(C\times P\). A thin open rectangle is a possible \(V\). Let \(\phi:V\to\Delta\) be a Riemann map. Choose \(0<\rho<1\) so that \(\phi(I)\Subset\Delta_\rho\). The map \[(\zeta,y,u)\longmapsto H(\phi^{-1}(\zeta),y,u)\] is defined near \(\overline\Delta_\rho\times C\times P\), with slightly smaller positive margins. Apply \({\cal A}_r\), using the \(\zeta\) disc as an extra passive factor. This gives an entire map \(E(\zeta,y,u)\) with arbitrarily good derivative approximation on a smaller product still containing \(\phi(I)\times C\times P\). Consequently \(E(\phi(s),y,u)\) approximates \(H(s,y,u)\) with the required derivatives along \(I\times C\times P\).

Choose a closed convex neighborhood \(I'\) of \(I\) contained in \(V\). Scalar polynomial approximation, with Cauchy estimates after a further harmless shrink, approximates \(\phi\) and any specified finite number of its derivatives on \(I'\) by a polynomial \(p\). At this stage \(E\) is fixed. On the compact sets in question all its derivatives up to the required order are bounded. The chain rule therefore shows that \(E(p(s),y,u)\) is as close to \(E(\phi(s),y,u)\) as desired. The map \(E(p(s),y,u)\) is entire in every variable. ◻

Proof. Write the variables as \((t,y,u)\), with \(t\) real on the face. Thicken \(F\) slightly within \(\mathbb R\times\mathbb C^{n-1}\), and thicken \(P\), so that all eventual estimates take place strictly inside the initial domain. Denote the thickened face again by \(F\). Its projection is a compact interval \(I\), and write \(F_t=\{y:(t,y)\in F\}\).

We first specify the finite cover used below. For each \(t_0\in I\), choose a compact convex neighborhood \(C_{t_0}\) of \(F_{t_0}\) in the \(y\) variables and an open complex disc \(D_{t_0}\) about \(t_0\) such that the given map \(f\) is defined near \[\overline D_{t_0}\times C_{t_0}\times P,\] including some fixed smaller margins. These choices are possible by compactness of \(F_{t_0}\times P\). After reducing the real interval about \(t_0\), it has \[F_t\Subset\operatorname{int}C_{t_0}\] uniformly on that interval. Indeed, otherwise a convergent sequence \((t_\nu,y_\nu)\in F\), with \(t_\nu\to t_0\), would contradict \(F_{t_0}\subset\operatorname{int}C_{t_0}\). Choose a finite subdivision of \(I\), with small overlaps at its division points, subordinate to these intervals. Denote the corresponding products by \(D_i\times C_i\times P\). All sets can be chosen a little larger than the sets used in the subsequent estimates.

At a division point \(b\), both adjacent \(C_i\)’s contain \(F_b\) in their interiors. Choose compact convex sets \(C^-\) and \(C\) with nonempty interiors such that \[F_b\subset\operatorname{int}C^-, \qquad C^-\Subset\operatorname{int}C, \qquad C\Subset\operatorname{int}(C_i\cap C_{i+1}).\] By the preceding compactness argument, after reducing a complex disc \(D_b\) about \(b\), the sets \(F_t\), for real \(t\) near \(b\), lie strictly inside \(C^-\), and \[\overline D_b\times C\times P \Subset (D_i\times\operatorname{int}C_i\times \operatorname{int}P^+) \cap(D_{i+1}\times\operatorname{int}C_{i+1}\times \operatorname{int}P^+)\] for an available larger passive polydisc \(P^+\). All transition intervals and their cutoff functions are fixed now. Also fix a large \(y\)-polydisc \(B\) containing all the \(C_i\) and all face projections, with a positive margin.

By \({\cal A}_{n-1}\), applied with the \(t\) disc and \(u\) polydisc passive, choose entire maps \(g_i(t,y,u)\) arbitrarily close to \(f\), with any fixed finite derivative accuracy, on \(\overline D_i\times C_i\times P\). More explicitly, apply the approximation assertion on slightly enlarged closed products still in the given domain and use Cauchy estimates on the displayed products. We join two adjacent \(g_i\)’s over one transition, with control on its required face slices.

On a slightly enlarged \(D_b\times C\times P\), a chart spray over \(f\), from Lemma 8, expresses the two maps as \[g_i=f^\sharp(\cdot,a_i),\qquad g_{i+1}=f^\sharp(\cdot,a_{i+1}),\] where \(a_i,a_{i+1}\) and their required derivatives are as small as desired. Choose a smooth function \(\chi:[0,1]\to[0,1]\) equal to \(0\) near \(0\) and \(1\) near \(1\), and set \[f_s=f^\sharp\bigl(\cdot,(1-\chi(s))a_i+\chi(s)a_{i+1}\bigr).\] This is a smooth \(s\)-family, holomorphic in \((t,y,u)\). It equals the entire map \(g_i\) near \(s=0\) and \(g_{i+1}\) near \(s=1\); on these subintervals use those entire definitions on all \(y\). It is arbitrarily close to \(f\) on the local product, and its positive \(s\)-derivatives up to any fixed order are arbitrarily small there. The constants here depend only on the previously fixed local chart and cutoffs.

The path \(f_s\) is defined only over the small convex set \(C\), except near its endpoints, where it equals an entire map. We need a path defined over the common large polydisc \(B\), still close to \(f_s\) on \(C^-\), and equal to the prescribed endpoint maps on all of \(B\). Direct interpolation in the chart over \(f\) cannot provide this, because that chart is not defined over \(B\).

We obtain the larger domain by expanding the \(y\) variable only where the path already equals an entire endpoint map. After entire approximation on the fixed small set \(C\), we undo that expansion. Choose \(d_*\in\operatorname{int}C\). For some \(L\geq1\), \[ d_*+L^{-1}(B-d_*)\Subset\operatorname{int}C. \tag{9}\] Choose a smooth positive function \(\lambda:[0,1]\to[1,L]\) which equals \(1\) outside the endpoint intervals where \(f_s\) is an entire endpoint map, and equals \(L\) on smaller endpoint intervals. All of these choices are fixed. The family \[\widetilde f_s(t,y,u) =f_s\bigl(t,d_*+\lambda(s)(y-d_*),u\bigr), \qquad y\in C,\] is defined on a common neighborhood of \([0,1]\times\overline D_b\times C\times P\). This assertion uses entirety precisely where \(\lambda>1\); where only the local definition of \(f_s\) is available, \(\lambda=1\). Apply Lemma 18 to make this family jointly holomorphic near that compact product, with arbitrarily good finite derivative accuracy. Next apply Lemma 19, with \({\cal A}_{n-1}\) and passive factors \(D_b,P\), to obtain an entire approximant \(E(s,t,y,u)\).

For real \(s\), put \[A_s(t,y,u)= E\bigl(s,t,d_*+\lambda(s)^{-1}(y-d_*),u\bigr).\] It is smooth in \(s\) and holomorphic in \((t,y,u)\) on the whole prescribed large product. On \(C'=C^-\Subset \operatorname{int}C\), which contains all required overlap slices, the inverse scaling stays in a fixed compact subset of \(\operatorname{int}C\): by convexity it lies in the convex hull of \(C'\cup\{d_*\}\). The two scalings cancel exactly: \[\widetilde f_s\bigl(t,d_*+\lambda(s)^{-1}(y-d_*),u\bigr) =f_s(t,y,u).\] Thus approximating \(\widetilde f_s\) on that fixed compact subset makes \(A_s\) arbitrarily close to \(f_s\), with the required derivatives, on these slices. On the smaller endpoint intervals, (9) gives the stronger statement that \(A_s\) is arbitrarily close to \(g_i\), respectively \(g_{i+1}\), on the entire large product \(\overline D_b\times B\times P\). The chain rule costs only finite constants, since \(\lambda\) and all scaling factors have already been fixed.

We may arrange exact endpoint joins. A chart spray over \(g_i\) exists on a slightly larger version of that large product. On a smaller initial \(s\)-interval, write \(A_s=g_i^\sharp(\cdot,\beta_s)\). Here \(\beta_s\), including all required derivatives, is as small as desired. Replace \(\beta_s\) by \(\kappa(s)\beta_s\), where \(\kappa\) is \(0\) near \(0\) and \(1\) before leaving this endpoint interval. Do the analogous replacement near \(1\). The repaired path, denoted \(H_s\), is exactly \(g_i\) near one endpoint and exactly \(g_{i+1}\) near the other, and retains the desired approximation on the overlap slices. It need only be holomorphic on a neighborhood of the fixed large product, which the chart construction provides.

There is no circular smallness requirement in this procedure. First choose the local \(g_i\)’s sufficiently accurate that the interpolated \(f_s\) has the desired small error after the fixed real transition cutoff. Then, with \(g_i,\lambda,B\) and the endpoint chart sprays fixed, choose the complexification and entire approximation of \(\widetilde f_s\) as accurate as needed to overcome all their finite derivative and chart constants. Finally perform the exact endpoint repairs.

Substitute a fixed smooth cutoff \(s=s(t)\) across the real transition interval. The result is a smooth real \(t\)-family, holomorphic in \((y,u)\) on the large product, which joins the two local entire maps and is as close to \(f\) as prescribed, with all required tangential derivatives, on the face slices. Doing this at the finitely many disjoint transition intervals gives one such family on all of \(I\). Extend it smoothly past the endpoints of \(I\) using the first and last entire maps \(g_i\), to which it already agrees on endpoint intervals.

Apply Lemma 18 once more, now to this real \(t\)-family on the fixed large polydisc in \((y,u)\). The result is jointly holomorphic near \(I\times B\times P\) and retains the prescribed accuracy on the face. Proposition 17 is \({\cal A}_0\); thus Lemma 19 with \(r=0\) converts this last family into an entire map in \((t,y,u)\), still with the required accuracy. When \(n=1\), there are no \(y\) variables and the scaling step is omitted; the same argument, or just this last complexification and \({\cal A}_0\) step, applies. ◻

Proof. Use several intermediate thickenings between \(Q_r\) and \(Q_{r-h}\), with successive distances fixed positive multiples of \(h\). Cover the required part of \(M_r\) by coordinate boxes of side comparable to \(h\). The enlarged boxes used below are still inside \(Q_r\). One can choose a grid cover with bounded overlap, at most \(C h^{-2n}\) boxes, and a smooth subordinate partition whose derivatives of order \(j\) are bounded by \(C_jh^{-j}\). Boxes outside a sufficiently large fixed ball are unnecessary.

In one enlarged box, with center \((t_0,z'_0)\) on the interface, take a smooth real cutoff \(\chi\) supported in its \(t\)-interval and equal to one on a smaller interval. For \(\operatorname{Im}\zeta\ne0\) put \[C_u(\zeta,z',v)=\frac{1}{2\pi i} \int_\mathbb R\frac{\chi(t)u(t,z',v)}{t-\zeta}\,dt.\] The one-variable Cauchy boundary formula gives the jump \(u\) on the smaller interval, after assigning the signs of the two boundary values in the usual way. Its Hölder bound follows by subtracting \(u(t_*,z',v)\) in the singular integral near a boundary point \(t_*\): the remaining numerator is bounded by a constant times \(|t-t_*|^\alpha\). The same subtraction in the difference of two boundary values gives the \(C^\alpha\) estimate. Rescaling the interval to unit length gives a bound with a fixed power of \(h^{-1}\). The integral is holomorphic in \((z',v)\) by dominated integration. Cauchy estimates on slightly smaller transverse boxes control transverse derivatives, with a further fixed power of \(h^{-1}\). Thus the local side functions have the asserted joint Hölder trace bounds.

Denote these local pairs by \(c_i^\pm\). Multiply by a smooth partition \(\theta_i\) in a band around the interface, equal in sum to one on a smaller band, and extend the partition terms into the two sides with support in their boxes. This gives smooth side functions \(f^\pm=\sum_i\theta_i c_i^\pm\) with trace jump \(u\). Their ordinary side derivatives \(\bar\partial f^\pm\) fit together to a single smooth form \(\eta\) across the interface. Here is the point that ensures smoothness despite the original Hölder data. In a smaller box choose one reference pair \(c_0^\pm\). Every difference \(c_i^\pm-c_0^\pm\) has equal traces, hence extends holomorphically across the cut by continuity and Morera’s theorem. Where \(\sum_i\theta_i=1\), \[\bar\partial f^\pm =\sum_i(\bar\partial\theta_i)(c_i^\pm-c_0^\pm).\] The right side is the restriction of a common smooth form. Outside this smaller band, the terms are evaluated a distance comparable to \(h\) from their Cauchy singularities. Consequently any fixed number of derivatives of \(\eta\) is bounded by \(C h^{-p_j}\|u\|_{C^\alpha}\). It is closed, since it is locally a \(\bar\partial\) derivative on the open sides and smooth across their boundary.

Solve \(\bar\partial w=\eta\) on an intermediate convex domain. A weighted minimal \(L^2\) solution with the fixed strictly plurisubharmonic weight \(|z|^2\) has \(L^2\) norm at most \(C\|\eta\|_{L^2}\) by the standard \(L^2\) theorem; the weight is bounded above and below on the domains in question; this is the weighted estimate of (Hörmander 1965, Theorem 2.2.1\('\)), also treated in (Hörmander 1990; Forstnerič 2017). Interior estimates give both a supremum and a first-derivative bound on the next smaller thickening, with polynomial losses in \(h^{-1}\). Indeed, writing \(\eta=\sum_{j=1}^n\eta_j\,d\bar z_j\), the distributional equation gives \(\Delta w=4\sum_j\partial\eta_j/\partial z_j\) componentwise. Interior elliptic estimates on nested balls of radius comparable to \(h\), followed by Sobolev embedding, bound \(w\) in \(C^1\) by its \(L^2\) norm and a fixed finite number of derivatives of \(\eta\), with powers of \(h^{-1}\). This proves the required interior bound and hence the Hölder trace bound for \(w\).

The operator just used is fixed and linear on the base domain. Applied to a holomorphic family of \(L^2\) coefficients, it gives a holomorphic family of \(L^2\) solutions. Interior estimates turn this into ordinary joint holomorphic dependence on \(v\). Now set \(a^\pm=f^\pm-w\). Their side \(\bar\partial\) derivatives vanish, their trace jump is unchanged, and the stated estimate follows by enlarging \(p\) to dominate the finitely many derivative and covering losses. ◻

Proof. Apply Lemma 21 to split the transition error as \(b-a=c\). With losses \(h\) in the base and fiber margins, its bound is \(Ch^{-p}e\), where \(e=\|c\|_{r,b,\alpha}\); here and below the side corrections are measured by their supremum and trace norms. Cauchy estimates in the fiber variable give the same bound, with additional fixed powers of \(h^{-1}\), for their first and second fiber derivatives on smaller balls.

As in Lemma 15, the corrected transition \(({\rm id}+b)^{-1}\circ\gamma\circ({\rm id}+a)={\rm id}+d\) is determined on the interface by \[d=c(v+a)-c(v)-\{b(v+d)-b(v)\}.\] The \(C^\alpha\) multiplication inequality and integration of the first fiber derivative along line segments bound the first difference by \(Ch^{-p}e^2\). To compare its values at two base points, the variation of the derivative at the shifted fiber argument is controlled by the second fiber derivative. The same estimates make the operation in braces contractive in the trace norm. Thus, after increasing the fixed exponent \(p\), \[\|d\|_{r-h,b-h,\alpha}\leq Ch^{-p}e^2.\] Only these finitely many derivatives are required at every step. The losses \(h_i=h_0 2^{-i}\) and the rescaled errors \(E_i\) in the proof of Lemma 15 consequently give the same doubly exponential convergence. The sums of side corrections and their fiber derivatives converge on both sides, including their \(C^\alpha\) traces. Their limiting transitions are the identity.

Composing the original sprays with the limiting side maps gives equal continuous traces. Near each interface point, continuity places both images in a common target chart; Morera’s theorem across the real hyperplane then proves holomorphicity of the joined map. The old-side approximation follows from the small fiber corrections and the fixed spray’s uniform continuity on compacta, exactly as in Lemma 15. ◻

Proof of Theorem 14. We prove \({\cal A}_n\) by induction. The case \(n=0\) is Proposition 17. Suppose \({\cal A}_{n-1}\) is known.

We first remove a single real affine constraint. After an invertible complex affine change of the \(n\) variables, write the enlarged convex compactum as \(Q\subset\mathbb C^n\), and the old compactum as \[Q_- = Q\cap\{\operatorname{Im}z_1\leq0\}.\] Assume a map \(f\) is given near \(Q_-\times P\). Cuts with no enlargement are trivial. In the nontrivial applications below, \(Q_-\) has nonempty interior and the cut meets the interior of \(Q\). Figure 2 shows the two sides to be joined.

Choose small convex thickenings of \(Q\) and of \(P\). Their lower sides, together with additional small neighborhoods, can be kept in the domain of \(f\). To verify this point without a transversality assumption on the boundary of \(Q\), note that as \(\eta\downarrow0\), \[(Q+\eta\overline B)\cap\{\operatorname{Im}z_1\leq0\} \longrightarrow Q_-\] in the following sufficient sense: every sequence of points in the left-hand sets has all its limit points in \(Q_-\). Compactness then places these sets in any prescribed neighborhood of \(Q_-\), for small \(\eta\). Here \(\overline B\) is the Euclidean closed unit ball.

Lemma 8 gives a chart spray \(f^\sharp(z,u,v)\) over a neighborhood of the thickened lower side, with a fixed auxiliary polydisc in \(v\). Its trace on \[\{\operatorname{Im}z_1=0\}\] is holomorphic near a convex compact face in \((t,y)=(\operatorname{Re}z_1,z_2,\ldots,z_n)\). Apply Lemma 20, treating \(u,v\) as passive polydisc variables. We obtain an entire map \(G(z,u,v)\) arbitrarily close to \(f^\sharp\), with, say, two tangential derivatives on a slightly enlarged face and with fixed smaller \(u,v\) margins.

On this interface, invert the vertical chart of \(f^\sharp\). For a fixed smaller \(v\)-polydisc this produces a transition \[\gamma(z,u,v)=v+c(z,u,v)\] between \(G\) and \(f^\sharp\), with \(c\) as small as desired in the \(C^\alpha\) trace norm, for any fixed \(0<\alpha<1\). This follows from the just obtained tangential derivative accuracy and the bounded derivatives of the fixed chart inverse on compact sets; Cauchy estimates on the retained \(u\)-margin also give the required Hölder control in \(u\). The transition is holomorphic in the complex tangential variables and in \(v\). Use \(f^\sharp\) on the lower side and \(G\) on the upper side. Apply Lemma 22 with convex base \(Q\times P\), normal coordinate \(z_1\), and only \(v\) as the fiber variable. With losses smaller than the chosen thickening margins, it glues their suitably reparametrized central maps on a neighborhood of \(Q\times P\). Its corrections can be made arbitrarily small on the lower side, so the resulting map approximates \(f\) on \(Q_-\times P\). No estimate for \(G\) away from the interface is used.

Finally let \(K\subset\mathbb C^n\) be any nonempty compact convex set, and let \(f\) be given near \(K\times P\). Thicken \(K\) and \(P\) slightly within this domain, and choose a full-dimensional real convex polytope \(K'\) containing \(K\) in its interior and still within the available thickening. Choose a large coordinate polydisc \(\overline\Delta_R^{\,n}\) containing \(K'\) in its interior. Write \[K'=\bigcap_{j=1}^M\{\ell_j(z)\leq b_j\},\] where the \(\ell_j\)’s are real linear forms, and remove these constraints one at a time inside \(\overline\Delta_R^{\,n}\). At each stage both compact sets are convex, the old one has nonempty interior, and a nonredundant constraint can be put in the form \(\operatorname{Im}z_1\leq0\) by a complex affine coordinate change. The single-constraint argument therefore extends the map approximately to the next compactum. There are only finitely many stages, so assign their errors a sum smaller than half the desired tolerance. The final map is defined near \(\overline\Delta_R^{\,n}\times P\), a polydisc. Proposition 17 approximates it by an entire map, using the remaining half of the tolerance. This proves \({\cal A}_n\), completes the induction, and gives Theorem 14 by taking no passive variables. ◻

Proof. Symplecticity and exactness respectively give \[\begin{align*} u_w+v_p&=-u_wv_p+u_pv_w,\tag{10}\\ 0&=\int_0^1 v(w,p)(1+u_w(w,p))\,dw. \tag{11}\end{align*}\] The second identity follows by expanding \((p+v)d(w+u)-p\,dw\) on a horizontal period circle; the integral of \(p u_w\) is zero. Consequently its mean \(v_0(p)\) satisfies \[|v_0(p)|\leq C\delta^{-1}e^2.\] Take the unique mean-zero periodic primitive satisfying \[H_w=-v+v_0.\] Integration along one horizontal period, followed by subtraction of the mean, bounds \(H\) by \(Ce\). This primitive is holomorphic in \(p\). Equation (10) gives \[\partial_w(u-H_p)=-u_wv_p+u_pv_w-v_0'(p).\] The right side has zero mean and is \(O(\delta^{-2}e^2)\) after one additional fixed fraction of the margin loss. Integrating its mean-zero primitive bounds \((u-H_p)-u_0\) by the same quantity. The other component of \(R\) is \(v_0\). Rescaling the allocation of \(\delta\) among these finitely many steps proves the stated estimate. ◻

Proof. Set \(T_j=-H_j\) and write \[T_j(w,p)=\sum_{n\neq0}t_{j,n}(p)e^{2\pi i n w}.\] The coefficients are holomorphic in \(p\). Denote the negative and positive parts by \(T_j^-\) and \(T_j^+\). Starting from \((w_j,p)\), continue the coordinates by \[w_{s+1}=w_s+a_s(p)\] in either direction, and define \[ G_j(w_j,p)=\sum_{l\geq j}T_l^-(w_l,p) -\sum_{l<j}T_l^+(w_l,p). \tag{13}\] For negative coefficients use a horizontal coefficient-bound line at height \(L_l+s-\delta/4\). An evaluation point of \(D_j(s-\delta,\rho)\) is below that line, in the continued \(l\)-th coordinate, by at least \[3\delta/4+(1-\kappa)(l-j) \qquad(l\geq j).\] For positive coefficients use height \(L_l-s+\delta/4\). The evaluation point is above that line by at least \[3\delta/4+(1-\kappa)(j-1-l) \qquad(l<j).\] For the nearest terms these inequalities are exactly the two defining inequalities for \(D_j\). For each further term an intervening block contributes at least \(1-\kappa\).

The absolute value of a Fourier mode is \(|e^{2\pi i n w}|=e^{-2\pi nY}\). Therefore both sums, including the sum over frequencies, are bounded by a constant times \[e\sum_{n\geq1}\sum_{q\geq0} \exp\bigl(-2\pi n(3\delta/4+(1-\kappa)q)\bigr) \leq C e\delta^{-1}.\] This proves normal convergence and holomorphicity, including in the parameter. Telescoping Equation (13) leaves \(T_j^-+T_j^+=-H_j\), proving Equation (12). Cauchy estimates on smaller \(D_j\) give the derivative bounds. In particular, one need not differentiate a sum of an unbounded number of translation functions term by term. ◻

Proof of Theorem 27. Start with slightly smaller \(s_0<h\) and \(\rho_0<r\). Fix a small \(d_0>0\) and take \(\delta_n=d_0 2^{-n}\). Choose \(d_0\) so that \[c_0\sum_{n\geq0}\delta_n <\tfrac14\min(s_0,\rho_0).\] At stage \(n\) reduce both margins by \(c_0\delta_n\). Thus their limits remain positive. Set \(B=C d_0^{-q}\), increasing the uniform constant in Equation (15) once if needed. If \(e_n\) bounds the normalized errors, then \[e_{n+1}\leq B2^{qn}e_n^2.\] For \(x_n=B2^{q(n+1)}e_n\) this becomes \(x_{n+1}\leq x_n^2\). Consequently, if \(x_0\leq\theta<1\), then \[e_n\leq B^{-1}2^{-q(n+1)}\theta^{2^n}.\] Every series \(\sum_n\delta_n^{-k}e_n\), with fixed \(k\), converges and tends to zero as \(e_0\to0\). Taking \(e_0\) still smaller ensures all the margin conditions above, keeps \(|\operatorname{Im}a_j^{(n)}+L_j|<h/3\), and keeps the translation derivative bound below \(A+1\). These estimates are uniform in \(j\).

The successive compositions of the \(K_j\) converge uniformly, with derivatives on smaller domains, to maps \(C_j\) into the original block domains. To see this directly, sum their displacements; their derivative norms are bounded by a convergent product of \(1+C\delta_n^{-3}e_n\). The reserved inclusions keep every composition in its domain. The limits remain symplectic because their differentials converge and each determinant is one. Put \(\widetilde\phi_j=\phi_j\circ C_j\) and \(\widetilde a_j=\lim_n a_j^{(n)}\). Passing to the limit in the chart identities on the seams gives \[\widetilde\phi_j(w,p)= \widetilde\phi_{j+1}(w+\widetilde a_j(p),p).\] The parameter radius has been decreased by the fixed sum of reserved losses and is still bounded below by a positive number depending only on the initial margins. A smaller common radius \(b\) can therefore be fixed before the error threshold, independently of the chain.

It remains to identify the domain, rather than invoke a uniformization theorem. Fix block zero, put \(w_0=w\), and define \(w_{j+1}=w_j+\widetilde a_j(p)\) in both directions. These are holomorphic shears, and \(dw_j\wedge dp=dw\wedge dp\). Use the \(j\)-th chart between the heights \[\operatorname{Im}w_{j-1}=L_{j-1},\qquad \operatorname{Im}w_j=L_j,\] including the surviving overlaps. In \(w_j\) coordinates the first height is near zero and the second is \(L_j\). Their separation is at least \(1-h/3\). Hence the successive edges tend to both infinities, locally uniformly in \(p\), and the charts cover all of \(\mathbb C\times\Delta_b\). They agree on overlaps by the limiting identity. Their common pullback form is \(dw\wedge dp\), proving immersion. Periodicity follows from the periodicity of the original charts and every correction. The summable estimates also prove the claimed approximation statements. ◻

Proof. For a transition \(F=(W,P)\) the closed one-form \(P\,dW-p\,dw\) has a constant complex period \[D=\int(P\,dW-p\,dw).\] It is independent of the representative period circle and of \(p\). Evaluation at an interior circle with \(p=0\), using Cauchy estimates on a fixed seam margin, bounds \(|D|\) by the error from a shear. Replace the added target chart by \((W,P_{\rm new})\mapsto\phi(W,P_{\rm new}+D)\). The transition now has \(P_{\rm new}=P-D\), so its period is \[D-D\int dW=0.\] The last integral is one because the transition has degree one. In a backward extension one instead translates the newly added input chart, with the corresponding sign. Uniform buffers permit these small translations.

Apply the identical substitution to the model chart. Its outgoing end is now the outgoing end of that updated model, and this is the position used to choose the following block. Both actual and model maps were defined on a larger disc, so their difference after restriction and translation has the same uniform bound. The old seam still has only a small error; the assertion does not require the two moved model centers to coincide there. Repeating this argument compares each new chart with its current model and never sums the previous matching errors. ◻

Proof. On a plain quadric, identify the two rulings with the projections of \(\mathbb P^1\times\mathbb P^1\). Their restrictions to \(E\) have degree two. They cannot both ramify at one point: otherwise the differential of the inclusion of the smooth curve \(E\) in the quadric would vanish. This proves the first assertion on \(S^+\).

For the blown-up quadric, a ruling fiber is additionally forbidden when it passes through a point of \(Z\). Choose \(Z\) generally so that its points avoid both ruling ramification sets, and so that a forbidden level of one ruling and a forbidden level of the other have a common point on \(E\) only at a point of \(Z\). To check that these are proper conditions, write the two projections as \(\pi_1,\pi_2\). Their ramification sets are disjoint. Conditions involving one ramification set and one \(\pi_i(Z)\) exclude only finitely many values for the corresponding point of \(Z\). For distinct \(z_i,z_j\), the condition that the grid point with coordinates \((\pi_1(z_i),\pi_2(z_j))\) lie on \(E\) is a proper condition on the pair. For \(i=j\) the grid point is \(z_i\) itself. Thus, away from \(Z\), at least one ruling remains regular.

Fix \(z_i\in Z\). Choose another center \(z_j\) and take the plane through the tangent line to \(E\) at \(z_i\) and through \(z_j\). For general choices its intersection with the quadric is a smooth \((1,1)\) curve. Its divisor on \(E\) is \[2z_i+z_j+z_k,\] with \(z_k\) different from \(z_i,z_j\), and it passes through no other point of \(Z\). Its strict transform is a member of the pencil of \((1,1)\) curves through \(z_i,z_j\). The chord through these two points is not a ruling line; hence the pencil has exactly these two simple base points, which are resolved by the blowups. An order-two contact at \(z_i\) becomes a simple transverse intersection with the strict transform of \(E\). The simple intersection at \(z_j\) disappears, and the intersection at \(z_k\) remains. The selected strict transform is therefore a regular fiber through the point of \(E\) over \(z_i\).

All the generic requirements just used are simultaneous finite proper conditions. More explicitly, on the elliptic curve the residual \(z_k\) is determined by the hyperplane divisor class and \(2z_i+z_j\); excluding coincidences or another center imposes a proper condition on at most three of the centers. The other requirements involve at most two centers and fixed branch data. The divisor \(Z\) ranges over the complete degree-sixteen linear system \(|\mathcal O_E(4)|\). Indeed the restriction of quartic forms is surjective, as follows from the Koszul resolution for the intersection of two quadrics. In an ordered general divisor of this degree any specified fewer than sixteen points vary freely; the remaining points satisfy the one elliptic sum condition. Hence the finitely many proper conditions above can all be avoided.

For each of the finitely many pencils, only finitely many members fail to be smooth or meet \(E\) nontransversely. Their union is a proper algebraic set, proving the second assertion. Finally, on any regular rational fiber, at least one of the two ruling projections is nonconstant. This is immediate for a ruling fiber by using the other ruling; for a \((1,1)\) fiber both projections are nonconstant. Away from finitely many points of the fiber the corresponding ruling member is regular and the projection has nonzero derivative. Such an intersection is transverse and can be chosen off \(E\). ◻

Proof. Over a small disc the smooth proper family of rational fibers is a holomorphic \(\mathbb P^1\) bundle and is trivial there. One can see this by choosing a local section, taking its relative degree-one line bundle, and using sections and base change to identify the family with a projective bundle. The two distinct end sections can then be put at zero and infinity.

In a parameter \(s\) on this disc write the logarithmic form as \(f(z,s)dz\wedge ds\). For each \(s\), the one-form \(f(z,s)dz\) on \(\mathbb P^1\) has only simple poles at zero and infinity. It is therefore \(b(s)dz/z\). Nondegeneracy makes \(b\) nonzero. Set \(dp=2\pi i b(s)ds\).

Let \(t_+(s)\) and \(t_-(s)\) be the exit and entrance positions on \(E\), and let \(\rho\) be the residue differential on this component. Taking residues gives the exact identities \[dp=2\pi i\,t_+^*\rho=-2\pi i\,t_-^*\rho.\] They also follow from the constancy of the elliptic sum of the two intersection points. Since both ends are transverse, \(t_+\) and \(t_-\) are locally invertible. Thus \(p\) moves each end freely and the stated residue interpretation follows. ◻

Proof. At an exit seam Equation (19) has leading part \(u=k_+(p)z\). Taking a local logarithm of the nonzero coefficient gives a change \(w_{\rm neck}=w_{\rm core}+a(p)\). Its derivative is bounded by the logarithmic derivative of \(k_+\). Its transverse change is exactly the identity at leading order: both parameters are primitives of the same signed residue differential, with the same constant at the matching nominal end.

At the entrance into the next component, the leading relation is \(v=k_-(p')/z'\) and \(uv=\alpha\). Hence \(z'\) is a nonzero holomorphic multiple of \(u\), so again \(w'=w+a(p)\), with the same orientation. The sign of the residue changes between the two components, and the use of \(z'=\infty\) at the entrance changes it once more. The transverse primitives therefore again agree as germs. The potentially large constant \(\log\alpha\) affects the translation itself, not its derivative in \(p\).

The actual seam maps differ from these leading maps by \(O(d)\) from the expansions and Equation (20), plus the core deformation error \(\epsilon(d,\alpha)\). These estimates are in \((\log u,t)\) or \((\log v,t)\) coordinates. The leading changes are uniformly invertible there. The inverse function theorem with the reserved buffers therefore produces the actual inverse branches throughout the seams. Local branches agree by uniqueness near the specified leading branch, including after going once around the period. Their lifts have degree one, and they are symplectic because both charts pull back the same surface form to \(dw\wedge dp\).

For completeness, the cuts can be prepared without imposing incompatible origins at the two ends of a block. Work first in raw coordinates. Enlarge each core range to \(cd<|z|<1/(cd)\) with a fixed sufficiently small \(c>0\), and add fixed logarithmic buffers to every neck. At a seam choose the two matching cut heights by its leading formula at the nominal center. Each raw block has independent room for the choice of its two cuts. Then translate its single long-coordinate origin so that its entrance is at height zero; its exit defines \(L_j\). For small \(d\), and then small \(\alpha\), all core and neck lengths are at least one. Shrink the fixed transverse discs so that the variation of \(\operatorname{Im}a_j\) about its central value is less than \(h/8\). The logarithmic derivative bounds show that this shrinking is independent of \(d\) and \(\alpha\). The two cut heights then give precisely \(|\operatorname{Im}a_j+L_j|<h/8\) with the required margins. ◻

Proof. We begin with an auxiliary elliptic K3 surface with section, given by the Weierstrass equation \[y^2=x^3+a(t),\] where \(a\) is a section of \(\mathcal O_{\mathbb P^1}(12)\) with twelve simple zeros, and the fundamental line bundle is \(\mathcal O_{\mathbb P^1}(2)\). The canonical bundle formula gives trivial canonical bundle, and \(R^1\pi_*\mathcal O=\mathcal O_{\mathbb P^1}(-2)\) gives irregularity zero. The simple zeros give cuspidal fibers with smooth total space. Thus the smooth Weierstrass surface is a K3 surface. These global Weierstrass and direct-image formulas are recalled in (Huybrechts 2016, chap. 11, Sections 1–2).

Let \(g\) be the automorphism which multiplies \(x\) by a primitive cube root of unity. Its fixed locus is the disjoint union of the zero section and the double cover \(x=0\) of the base. The latter is branched at twelve points and has genus five. The fixed locus has Euler characteristic \(2+(2-10)=-6\). The topological Lefschetz formula, in the order-three K3 formulation of (Artebani and Sarti 2008, proof of Theorem 2.2), gives \[\operatorname{tr}(g\mid H^2)=-8.\] If \(r\) is the invariant rank, the other eigenvalues occur in conjugate pairs of primitive cube roots, so \(r-(22-r)/2=-8\), whence \(r=2\). The fiber and zero section generate a unimodular hyperbolic plane in this invariant lattice; consequently the invariant lattice is exactly \(U\). Its orthogonal complement is even unimodular of signature \((2,18)\) and contains two hyperbolic planes.

That complement contains a primitive vector of square \(2d\). Eichler’s criterion, applied to the unimodular K3 lattice with its two hyperbolic summands, says that primitive vectors with the same square are isometric (Gritsenko et al. 2009). We may therefore transport the preceding lattice isometry so that the specified vector \(v\) belongs to its noninvariant complement. There \(1+g+g^2=0\), and thus \[(v,v)=(gv,gv)=2d,\qquad (v,gv)=-d.\] The plane \(P_0=\operatorname{span}_{\mathbb R}\{v,gv\}\) is positive. Choose either of its two orientations, and realize it as a marked K3 period by period surjectivity. The invariant \(U\) is algebraic at this period and contains a positive integral class, so the resulting surface is projective.

Choose a primitive isotropic generator \(e\) of this \(U\), with sign chosen so that its represented class is effective. We keep the abstract lattice data \(g,U,e\) fixed and change the marking by reflections in algebraic \((-2)\)-classes so that the class represented by \(e\) becomes nef. The reflections fix \(P_0\) pointwise, hence fix \(v\); all preceding lattice relations are unchanged. For completeness, if an effective isotropic class has negative intersection with an irreducible curve, that curve is a \((-2)\)-curve; reflection in it reduces the positive integral degree against a fixed ample class. Repetition terminates in the nef cone. The primitive nef isotropic pencil theorem then gives a base-point-free complete pencil \(|e|\) with \[h^0(\mathcal O(e))=2,\qquad H^1(\mathcal O(e))=0.\] It defines a genus-one fibration on our center.

We must show that this particular pencil is isotrivial. The \(g\)-invariant period planes of the chosen eigenvalue form a complex ball: they are the positive lines in one of the two nonreal eigenspaces of \(g\) on \(U^\perp\otimes\mathbb C\). Take a small ball through \(P_0\), realized by a marked local K3 family. At a generic point its algebraic lattice is precisely \(U\). To see this, each integral class outside \(U\) imposes a proper linear Hodge condition; such a condition cannot vanish on the entire eigenspace because the two conjugate eigenspaces span \(U^\perp\otimes\mathbb C\). Their countable union has empty interior.

The line bundle with class \(e\) extends to this small family. One way to verify the extension is to take a contractible Stein base. The topological class extends to the total space and has zero image in \(H^2(\mathcal O)\): by the Leray spectral sequence and Stein vanishing this group is the space of sections of \(R^2\pi_*\mathcal O\), and the class vanishes fiberwise because it is of type \((1,1)\). The exponential sequence supplies the line bundle. The vanishing of \(H^1(\mathcal O(e))\) and cohomology and base change extend its two sections. Their having no common zero is open by properness. Thus the genus-one pencils persist near \(P_0\).

At every nearby generic period, \(g\) acts identically on the Picard lattice, hence fixes an ample class. Global Torelli realizes it as an automorphism of order three. Suppose its induced action on the pencil base were nontrivial. All its fixed points would then lie over the two fixed points of that base action. Every fixed curve would be vertical. Since an irreducible vertical curve \(C\) satisfies \((C,e)=0\), the Hodge index theorem gives \(C^2\leq0\) and adjunction gives \(p_a(C)\leq1\). The fixed locus of a finite-order automorphism of a smooth surface is a disjoint union of smooth curves and isolated points. It would therefore have nonnegative Euler characteristic, contrary to the value \(-6\) determined by the cohomological action.

The base action is consequently trivial. The multiplier on a smooth fiber’s holomorphic differential equals the primitive cube-root multiplier on the surface form. A smooth elliptic curve with such an automorphism has \(j=0\). Finally fix any smooth member of the central pencil. It remains smooth in the extended pencil over a neighborhood in the deformation base, and its \(j\)-invariant varies continuously there. The nearby generic values are zero, so the central value is zero as well. This proves the assertion for every smooth member at \(P_0\). The relation \((e,v)=0\) holds because \(e\) belongs to the original invariant lattice. ◻

Proof. Integration over the elliptic fibers in local product charts expresses the pullback symplectic volume as a fixed positive constant times \(i\eta_\gamma\wedge\overline{\eta_\gamma}\). The base change has finite degree and \(X_0\) is compact, so this one-form has finite \(L^2\) norm on \(B^\circ\). In a punctured local coordinate, a holomorphic one-form with finite \(L^2\) norm has no negative Laurent coefficients. It thus extends holomorphically across each puncture. A local primitive of the extended form also extends the circle-valued action function.

The holomorphic forms \(\eta_{\gamma_1}\) and \(\eta_{\gamma_2}\) have constant nonreal ratio. Consequently an invertible real linear map identifies the pair of their imaginary parts with a complex-linear differential. Endow \((\mathbb R/\mathbb Z)^2\) with this constant complex structure. Then \(h\) is a nonconstant holomorphic map from the compact Riemann surface \(B\) to an elliptic curve. It has finite fibers.

A point without two safe choices satisfies two forbidden-level conditions. Each pair of the integral linear functions \(h_{\gamma_1},h_{\gamma_2},h_{\gamma_1}+h_{\gamma_2}\) is an automorphism of the real two-torus. Thus any two such conditions restrict \(h\) to a finite set of torus values. The inverse image is finite by the preceding paragraph. ◻

Proof. The centers may move after each exactness correction. We therefore prepare the compact region in which the construction will continue before choosing the charts. Let \(\rho>0\) be a common lower bound for the distance of the specified initial actions from their forbidden sets. For every cycle used in the initial data, put \[K_\gamma=\{b\in B: \operatorname{dist}(h_\gamma(b),F_\gamma)\geq\rho/4\}.\] This set is compact in \(B^\circ\). Choose finitely many product-torus charts covering it, with larger buffered domains still lying in \(B^\circ\). All subsequent restrictions and recentering will use this finite atlas. We shall prove that the updated centers remain a fixed distance inside the prepared region.

In a chart for a chosen cycle, the torus has constant lattice \(\mathbb Z+\tau\mathbb Z\), where \(\operatorname{Im}\tau>0\) after choosing a positive complementary cycle. Unroll this complementary period while retaining \(w\) modulo \(\mathbb Z\). We obtain finite cylinders of any prescribed fixed length. Starting sections can be chosen at any fiber phase, so the chart families cover all phases, not just a particular section.

At matching nominal base positions, successive model cylinders are identified on their overlaps by \[ (w,p)\longmapsto(w+a(p),p). \tag{24}\] The functions \(a\) incorporate both changes of local section and any required number of complementary periods. The latter are constant, because the modulus is constant. On smaller members of the finite buffered atlas, the derivatives of the remaining translation functions have uniform bounds. Choose long block cores containing the prescribed cells and leave several period cells as buffers at each end. At a joint, integral complementary translations let the two cut heights agree at the center with a bounded adjustment inside these buffers. Both cuts of each raw chart may be chosen independently. After translating the long origins, these charts have the block and overlap geometry required by Theorem 27. Shrinking the transverse radius makes the variations of their cut heights uniformly smaller than the overlap margins.

Each cylinder times its transverse disc is Stein. The local deformation argument used earlier therefore deforms these maps on their finite buffered domains to the nearby fibers. Although a cylinder map is usually not injective, it is an immersion, and on each overlap its local inverse branch close to (24) is unique. The branches consequently agree around the cylinder, including its period identification. The resulting transition is a degree-one periodic lift. The pulled-back forms can be normalized to \(dw\wedge dp\) by the transverse integration used in Theorem 27. All these constructions are uniform on the finite atlas, and their errors tend to zero with \(t\).

We next compare the action of an individual deformed block with that of its model, before any blocks have been joined. Let \(b\) be the nominal base position of a freshly centered block, and let \(l_b\) be the image of a period loop on its zero slice. Uniformly over the prepared charts and their allowed recenterings, \[ \operatorname{dist}_{\mathbb R/\mathbb Z} \bigl(h_\gamma(b),H_{\sigma_t}(l_b)\bigr)<\varepsilon_t, \qquad \varepsilon_t\longrightarrow0. \tag{25}\] To verify this, under the fixed smooth identification the real forms \(\operatorname{Im}\sigma_t-\operatorname{Im}\sigma_0\) are exact and tend to zero smoothly. Fix a Riemannian metric on the underlying compact manifold and let \(G\) be its Hodge Green operator. For the exact two-form \(\eta_t=\operatorname{Im}\sigma_t-\operatorname{Im}\sigma_0\), put \(\beta_t=d^*G\eta_t\). Its harmonic projection vanishes and \(dG\eta_t=Gd\eta_t=0\), so Hodge decomposition gives \(d\beta_t=\eta_t\). Elliptic regularity makes \(\beta_t\) tend to zero smoothly; see (Chen and Li 2017, Theorem 3.2 and Equation (3.26)) for this Hodge decomposition and Green-operator formulation. Integration of these primitives around the model loops, together with the small cylinders between nearby model and actual loops, proves (25). The loop families are compact: the base positions and fiber phases range over compact sets, and the number of wraps in a block is fixed. The estimate depends on the comparison of this block with its own model, not on any previous choices of centers.

Starting from any chosen block, extend the chain in both directions. At a joint initially use a chart with matching nominal base position. Its actual symplectic transition is close to (24). Its period defect is the constant \[\int_{\mathbb R/\mathbb Z}(P\,dW-p\,dw).\] Moving this candidate’s transverse center by the appropriately signed defect makes the period exactly zero. Retain this translated actual chart as the next block, so the exact overlap just obtained is preserved. Move its nominal center by the same amount in the model \(p\) coordinate; that updated position is used only when selecting the following candidate block. The adjustment is small, and comparison with the model of the newly centered chart retains its original error bound. It remains to justify that the nominal positions never approach a puncture.

For this purpose use the action (23) on the actual surface. Its value on the period loop of the zero slice is constant within a block. It is also constant between adjacent blocks. Indeed the primitive of \(dw\wedge dp\) is \(-p\,dw\); the change of action between the image of a zero-slice loop and the next zero-slice loop is the imaginary part, up to sign, of the displayed period defect. The image loop winds once in the next cylinder. Vanishing of the defect proves equality of its action with that of the next zero slice. Denote this common circle value by \(c\).

Thus the single-block estimate (25) reads \(\operatorname{dist}_{\mathbb R/\mathbb Z}(h_\gamma(b),c)<\varepsilon_t\) at every stage. We can now verify that the next center stays in the prepared region. Require the action error and the change in nominal action caused by a single center adjustment to be less than \(\rho/8\). For an initial block the conserved value \(c\) has distance greater than \(7\rho/8\) from \(F_\gamma\). Inductively, a next nominal center may first be chosen at the preceding one; the small adjustment keeps it inside the prepared region. The exact action comparison then improves its distance from the forbidden set to greater than \(3\rho/4\). This restores a fixed margin for the following step. The same argument applies backwards. Thus the entire doubly infinite chain is available with common transverse and overlap margins.

We have obtained exact symplectic transitions with uniformly small errors and the block geometry required by Theorem 27. That theorem supplies the periodic strip and arbitrarily small corrections on every prescribed compact part of the initial block. Estimates on slightly larger domains give the asserted \(C^1\) closeness. The transverse width has a uniform positive lower bound after the fixed margin losses. ◻

Proof. Fix such a point \(x\) and take one leaf of each strip through it, using sheets whose preimages lie inside the buffered initial blocks. Identify a smooth tube of the central torus with a disc times that torus, and let \(q\) be its smooth projection to the torus. On a sufficiently large finite box in the universal plane cover of the torus, the lifted projections of these two leaf parametrizations are \(C^1\) close to invertible affine maps. The prescribed cylinder lengths may be chosen to include this box and every lattice translate used below, with a fixed positive buffer. These choices are made on the central model; the required \(C^1\) tolerance is then fixed on these finite buffered domains. A small \(C^1\) perturbation of an invertible affine map is injective on a slightly smaller convex box, and its image contains a further smaller box. We may therefore invert the two projections and express the two leaf pieces as graphs over one common convex box \(Q\) in the torus cover. Choose this box to contain a fundamental parallelogram for the two cycles and buffers beyond every edge. The finite block lengths were chosen large enough to accommodate these translates and buffers.

Each graph is periodic under its own primitive cycle wherever both points and the required buffers belong to \(Q\). To check this exact assertion, shifting \(w\) by \(1\) leaves its strip map unchanged. The lift of its torus projection changes by an integral lattice vector; closeness to the original chart forces that vector to be exactly the chosen cycle. Inversion and uniqueness in the buffered box give the claimed graph equality.

Suppose the two holomorphic curve germs at \(x\) were equal. The set where the graph germs agree is open in \(Q\). It is also closed. At a limit of such points, both graph images have the same limiting point. In small immersion charts at that point, the two embedded holomorphic curve germs have accumulating intersections, so the identity theorem makes them equal. The use of the smooth projection causes no difficulty here: it only supplies graph coordinates, while the identity theorem is applied in holomorphic surface charts. Connectedness of \(Q\) now makes the graphs agree throughout \(Q\).

The common graph is periodic under both edge identifications. It consequently defines a compact smooth torus mapped into the tube whose local image is a holomorphic curve. Let \(m\) be the index of the sublattice generated by the two cycles in the full period lattice. Contraction of the disc component shows that the represented integral homology class in \(X_t\) is \(m e\), with \(m>0\); in the application from Lemma 37, \(m=1\). The pullback of \(\sigma_t\) to a holomorphic curve is zero, contradicting (26). The two leaf germs are therefore distinct.

They need not be transverse at \(x\). Take a local foliation chart for the second strip. Its transverse coordinate restricts to a nonconstant holomorphic function on the first leaf, since the two germs are distinct. Its derivative is nonzero at points arbitrarily close to \(x\). At such a point the first leaf meets a member of the second strip family transversely. Apply Remark 7 at this intersection, assigning the first strip, whose leaf contains \(x\), to the role of \(\sigma_2\) and the transverse partner to the role of \(\sigma_1\). Proposition 4 then gives \(L\) at \(x\) by the permitted displacement along the \(\sigma_2\) leaf. ◻

Proof of Theorem 35. By Lemma 37, all central fibers except finitely many have two independent safe cycle choices. Let \(A\subset X_0\) be the union of the excluded fibers and the singular fibers. This is a proper analytic subset of dimension at most one.

We make a finite compact preparation before deforming. At every point of \(X_0\setminus A\), choose smaller regular torus patches with two safe cycles and larger buffered patches. At a point of \(A\), choose a small holomorphic coordinate line disc through the point whose boundary avoids \(A\). Such a disc exists because a generic line is not contained in \(A\) and meets it discretely. Two sufficiently small different tilts give transverse discs with boundaries still outside \(A\). Their centers may vary in a smaller coordinate neighborhood. Compactness of \(X_0\) permits a finite choice of these center neighborhoods and regular patches. Enlarge the compact regular data to include all the disc boundaries. Every selected safe action then has a positive common margin from its finite forbidden set.

Apply Lemma 38 to these finitely prepared data, with blocks long enough for Lemma 39. On all sufficiently small deformations in \(\mathcal D_v\), the initial charts still cover fixed smaller tube patches, and the coordinate-disc boundaries remain in those patches. If (26) holds, Lemma 39 gives \(L\) at every point of the required regular patches. Corollary 12 gives \(L\) in the reserved center neighborhoods. Thus \(X_t\) has \(L\) everywhere.

Finally (26) holds on a nonempty relatively open subset arbitrarily close to \(P_0\) in \(\mathcal D_v\). Indeed \((e,v)=0\), but the functional \((e,\cdot)\) does not vanish identically on the positive vectors of \(v^\perp\). Varying the other normalized positive direction therefore breaks the equation \((e,[\sigma_t])=0\). Local Torelli identifies our small deformation neighborhood with a period neighborhood, giving the desired nonempty relatively open set \(\mathcal O_v\). For any other marked K3 with the same period, global Torelli after matching Kähler chambers identifies the underlying surface, so the property has the stated period-theoretic scope. ◻

Proof. Theorem 1 and the convex approximation characterization (Forstnerič 2006, Theorem 0.1) imply that \(S\) is Oka. Every Oka manifold is Oka-1, which is precisely the approximation and interpolation property stated here. The CAP-to-interpolation implication is established in (Forstnerič 2005, Corollary 1.3); for the Oka-1 formulation with individually prescribed jet orders, see (Alarcón and Forstnerič 2025, Definition 1.1 and the following discussion). Since \(\dim_{\mathbb C}S=2\), the same interpolating map can be chosen to be an immersion by (Alarcón and Forstnerič 2025, Corollary 2.10) when all prescribed first derivatives are nonzero. ◻

Proof of Corollary 2. Choose a countable dense sequence \((x_j)_{j\geq1}\) in the ordinary topology of \(S\), set \(x_0=x\), and choose nonzero vectors \(v_j\in T_{x_j}S\), with \(v_0=v\). At the distinct source points \(a_j=3j\), choose holomorphic germs \(g_j:(\mathbb C,a_j)\to(S,x_j)\) with \(g'_j(a_j)=v_j\). Explicitly, if \(\kappa_j\) is a chart taking \(x_j\) to the origin in a ball in \(\mathbb C^2\), put \[g_j(z)=\kappa_j^{-1}\bigl((z-a_j)d\kappa_j(v_j)\bigr).\] Choose \(0<r_j<1/3\) so that this formula is defined near \(\{|z-a_j|\leq3r_j\}\). These closed discs are disjoint and locally finite.

The germs are restrictions of a single continuous map \(h:\mathbb C\to S\). To construct it, fix \(b\in S\) and a continuous path from each \(x_j\) to \(b\). Use \(g_j\) on \(|z-a_j|\leq r_j\). On the annulus \(r_j\leq|z-a_j|\leq2r_j\), multiply \(\kappa_j(g_j(z))\) by a continuous radial cutoff equal to one near its inner boundary and zero near its outer boundary. Convexity of the coordinate ball keeps the resulting map in the chart. On \(2r_j\leq|z-a_j|\leq3r_j\), follow the chosen path from \(x_j\) to \(b\) with a radial parameter constant near the two boundaries. Put \(h=b\) elsewhere. Local finiteness and the boundary agreements prove continuity; \(h\) is holomorphic near every \(a_j\) and near the compact Runge disc \(K=\{|z|\leq r_0/2\}\).

Apply Corollary 41 to \(h\), \(K\), and the closed discrete sequence \((a_j)_{j\geq0}\), prescribing the first jet at every \(a_j\). Since every \(v_j\) is nonzero, this gives an interpolating holomorphic immersion \(f:\mathbb C\to S\). Thus \[f(a_j)=x_j,\qquad f'(a_j)=v_j\quad(j\geq0).\] The case \(j=0\) gives the prescribed point and exact tangent vector. The image contains the dense sequence \((x_j)_{j\geq1}\), proving ordinary density. If \(S\) is projective, its Zariski closed subsets are closed in the ordinary complex topology, proving the last assertion in that case. ◻

Proof. Every complex Enriques surface has an unramified holomorphic K3 double cover (Gallego et al. 2006, Introduction). Its covering surface is Oka by Theorem 1, and the Oka property descends through holomorphic covering maps (Forstnerič and Lárusson 2014, Introduction). This proves the Enriques assertion. Apart from K3 and Enriques surfaces, the minimal compact complex surfaces of Kodaira dimension zero are complex tori, bielliptic surfaces, and Kodaira surfaces, which are Oka by the established cases recalled in (Forstnerič and Lárusson 2014, Introduction).

A minimal class-VII surface has \(b_1=1\) and \(\kappa=-\infty\). When \(b_2>0\), the global spherical shell theorem (OpenAI 2026, Theorem 1.1) therefore applies, so the surface is a Kato surface. The class-VII decomposition recalled in (Forstnerič and Lárusson 2014, Introduction) now makes the list exhaustive: Hopf and Inoue surfaces for \(b_2=0\), and Enoki, Inoue–Hirzebruch, and intermediate surfaces for \(b_2>0\). Theorem 4 of (Forstnerič and Lárusson 2014) shows that exactly the Hopf and Enoki cases are Oka. ◻

Proof. Projective complex K3 surfaces are Oka by Theorem 1. Complex Enriques surfaces are projective (Gallego et al. 2006, sec. 2) and are Oka by Corollary 42. The conclusion follows from Forstnerič and Lárusson’s theorem that every projective Oka manifold is elliptic (Forstnerič and Lárusson 2026, Theorem 1.1). ◻