Pointed flat piecewise linear caps admit saddle-free replacements

Extend $j$ to a PL homeomorphism $f:Z\to Z_0$. Use the supplied parameter of $Z$ to transfer $j$ to the boundary of a convex ball, triangulate it there and on $\mathrm{\partial }{Z}_0$, and cone the boundary map from interior points. Convexity makes the matching cone-simplex maps bijective with PL inverses. Choose a common finite triangulation on which $f,h,h_0\circ f$ are linear. The last two functions have exactly the same zero vertices, namely those on $D$, and are strictly positive at all other vertices. Cut at a number $\delta >0$ below all their positive vertex values and below the top height. On corresponding simplices the two cuts have identical face posets: they retain the zero vertices and cut precisely the zero-positive edges. Inductively match the truncated cells, fixing $j$ on their old boundary faces, matching the top sections by compatible face maps, and coning the remaining boundaries between interior points at height $\delta \mathrm{/}2$. Such points exist on every mixed face by convexity. The map preserves height on every cone simplex. This gives a PL homeomorphism $f_{\delta }:Z\cap \{h\le \delta \}\to Z_0\cap \{h_0\le \delta \}$ preserving height and equal to $j$ on the old boundary. This is the explicit zero-vertex collar-leveling argument; it does not assert that the original extension $f$ preserved height.

Choose an affine function $L$ with $L(Q_0)=0$, $L>0$ on $D_0\setminus \{Q_0\}$, and $L\ge 0$ on $D_0$: the sum of two supporting inequalities of the polygon at its vertex does this. For sufficiently small $e>0$, $h_0>eL$ on $G_0\setminus \{Q_0\}$ and on the section $h_0=\delta$. The first assertion follows by checking the two nonzero vertices of the triangle and using affinity; the second follows from compactness and $\delta >0$. Then $W_0=Z_0\cap \{h_0\le \delta ,h_0\le eL\}$ is a convex three-ball. Its new face $A_0^{\mathrm{\prime }}=Z_0\cap \{h_0=eL\}$ is a convex polygonal disk lying below $\delta$, and its remaining boundary $A_0$ lies in the old boundary of $Z_0$. It contains $D_0$, meets $G_0$ exactly at $Q_0$, and misses the top section. Both boundary disks have finite PL parameters: $A_0^{\mathrm{\prime }}$ is convex, and the complement of the relative interior of a face in a convex polyhedral sphere is a PL disk, as follows by the finite truncated-vertex planar chart and polygonal disk-extension construction. In particular $\mathrm{\partial }{W}_0=A_0\cup A_0^{\mathrm{\prime }}$ and their intersection is their common boundary.

The retained body $Z_0\cap \{h_0\ge eL\}$ is convex and has its unique height-zero point at $Q_0$. Every new point of its boundary away from $Q_0$ and from the unchanged attachment boundary is regular for height: at a new vertex strictly between the extrema, its convex link circle has a positive arc and a negative arc, each connected, because its lower and upper cone sections are convex; face and edge charts give the same conclusion. Choose $L$ and then $e$ outside the finitely many exceptional values making the new planar face horizontal or introducing a non-extremal horizontal face. This retains the strict inequalities in step 2.1. Coning the link arcs gives the local height-product charts, equivalently by the explicit two-half-space polygonal fan straightening. Pull $W_0,A_0,A_0^{\mathrm{\prime }}$ back by $f_{\delta }$. Height preservation gives the claimed new regular charts and the single zero point $Q$. The disk replacement is confined below any initially prescribed positive bound by taking $\delta$ smaller.

Let $E$ be the rest of the sphere after deleting $\mathrm{Int}A$. Its intersection with the pulled-back ball is $\mathrm{\partial }A$ by the stated transfer hypothesis. Thus $E,A,A^{\mathrm{\prime }}$ satisfy [F2]. The supplied ball bounded by $A\cup A^{\mathrm{\prime }}$ and any supplied ball bounded by $E\cup A^{\mathrm{\prime }}$ give a supplied ball bounded by $E\cup A$. All triangulations, cuts, cones and exceptional-value exclusions used finite collections; no choice assumption is introduced.