Smooth proper disks have PL approximations in a compatible triangulation

After a small smooth perturbation, keep $e$ proper and transverse to $\mathrm{\partial }M$ and make it transverse to the finite strata of $h(T)$, avoiding the vertices. Here the elementary finite parameter argument suffices: in each of finitely many ambient chart cores add small coordinate translations multiplied by a bump equal to one on the core. Evaluation on that core is a submersion in the translation parameters, so the regular values of the projection of the corresponding inverse-image stratum give transverse perturbations by [F2]. At boundary points use the tangential chart translations on $\mathrm{\partial }M$ and their product extensions in its collar. Choose the finitely many perturbations successively small enough to retain already obtained transversality and the disk's embedding and proper collar. Compact transverse intersections with the edge strata are finite; intersections with face interiors are compact regular arcs and circles after adding those edge endpoints, hence a finite embedded graph. There are no intersections with vertex strata.

Cut the disk along the finite inverse-image graph. Regard repeated slit sides as separate boundary copies before regluing. The pieces are compact smooth surfaces with corners: face-intersection arcs have nonzero differential; at an edge intersection their incident tangent directions are distinct, since the disk plane is transverse to that edge, so the two bounding equations in each sector give a corner chart by the inverse theorem. Supply each common boundary arc with one smooth regular parametrization, subdivided identically on its two copies. These are admissible boundary data in [F2], because the two corner rays are independent. Apply its finite relative whole-star construction to the pieces and reglue their identical boundary subdivisions. This gives a finite piecewise smooth whole-star triangulation of the disk with each simplex mapping into one ambient simplex. The inverse of $h$ on that simplex is smooth up to its indicated faces by the full-rank extension and inverse theorem. Consequently $g=h^{-1}e$ on this finite source complex is piecewise smooth. Its differential fan is injective: the ambient differential fan of $h$ is a homeomorphism onto the tangent half-space or space, and the disk differential is injective, so taking its inverse on the subdivided tangent sectors is injective. Thus $g$ is a whole-star embedding into a Euclidean realization of $\mathrm{\mid }T\mathrm{\mid }$.

Apply the shape-controlled secants of [F1] to $g$. Every refined simplex maps by affine interpolation of its vertex values into the same convex ambient simplex as its original carrier. Adjacent source simplices have identical interpolants on their common face. Thus these secants define a PL map into $\mathrm{\mid }T\mathrm{\mid }$, and sufficiently small strong error makes it an embedding by [F1]. They send boundary into the boundary subcomplex, since all the vertices of a boundary source simplex lie in its one boundary ambient face and the interpolant preserves that face.

Properness is retained as well. In a smooth disk collar the inward boundary defining coordinate of $M$ has strictly positive inward derivative, uniformly after a finite shrink, by the given transversality. Cellulate that collar as strips with its boundary as the only zero-coordinate face. In each ambient half-space simplex the corresponding affine boundary coordinate is nonnegative and positive on the vertices off that boundary face. Its affine interpolant is therefore zero precisely on the source boundary face. The derivative lower bound ensures these are precisely the collar sectors, even after a finer shape-controlled subdivision. Away from the collar the compact disk has a positive distance from $\mathrm{\partial }M$, and a sufficiently small value error retains that separation. Hence the PL disk meets $\mathrm{\partial }M$ exactly in its own boundary.

Along the original boundary, the secant curve lies in an arbitrarily small annular neighborhood and is transverse to its interval fibres: its tangential derivative is close to the nonzero derivative of the original parametrized curve in each of the finitely many sectors. Projection to the original circle is therefore a degree-one local homeomorphism, hence a homeomorphism. The new boundary is a graph in that annulus; interpolating its graph coordinate with zero gives an isotopy to the original boundary. Thus its nullhomotopy class, including its essentiality, is unchanged. The image under $h$ of the secant disk is the required properly embedded PL disk in the supplied compatible structure.