Delaunay-seeded curved uniform NVB refinement

Input the exact dyadic square $[-1,1{]}^2$, a tolerance $0<\tau \le 1\mathrm{/}4$, and the exact shear $\mathrm{\Phi }(x,y)=(x,y+x^2\mathrm{/}4)$. Use the unweighted indexed corner list $S=((1,(-1,-1)),(2,(1,-1)),(3,(1,1)),(4,(-1,1)))$ with all weights absent, and call Incremental planar Delaunay insertion with exact edge flips in its ordinary $g=0$ branch with no prescribed insertion permutation, so sites are inserted in increasing identifier order. If the call does not return a triangulation, return unresolved. Verify that the output consists of two triangles covering the square and sharing one diagonal; otherwise return unresolved. The boundary features are the four closed edges and four vertices; features are incident when their sets intersect, and separation is the minimum Euclidean distance between nonincident feature sets. Record the interior boundary-angle certificate $\alpha =\pi \mathrm{/}2$ and the nonincident-feature separation certificate $\sigma =1$. Label the shared diagonal reference on both triangles and label the boundary edges nonreference. For each seed triangle, sort its three input site identifiers increasingly and use the resulting triple as its seed key. Sort the two distinct keys lexicographically and assign operational root ranks $1,2$ in that order. Initialize their identifiers $(1,\mathrm{\varnothing }),(2,\mathrm{\varnothing })$ and stored vertex lists by Compatible 2D newest-vertex-bisection meshes, retaining the just-assigned compatible reference-edge labels. All subsequent calls use the resulting persistent root-and-binary-path order, including on generated children; the site-key order is bookkeeping and does not relabel an edge. Set $k=0$. While $(3\mathrm{/}2)2^{-k}(2\sqrt{2})>\tau$, set ${\eta }_K^2=1$, $o_K^2=0$ on every current cell and call Dorfler marking and conforming newest-vertex refinement with ${\theta }_E=1$, ${\theta }_O=1$. Start the cycle budget at $B=1$; if the call returns budget-exhausted, retry from the unchanged mesh with $B$ doubled until it returns complete. A complete call requests three generations on every cell and uses recursive conformity closure; then set $k\leftarrow k+1$ and test the diameter bound again. Map every output reference triangle by the common shear $\mathrm{\Phi }$ and return the mapped cells, feature certificates, and $k$. If a predicate is unresolved, the Delaunay seed does not have the stated two-triangle form, or closure returns invalid input, return unresolved.

The exact Delaunay seed uses Planar triangulations, Delaunay structures, and Voronoi cells and Incremental planar Delaunay insertion with exact edge flips. NVB means newest-vertex bisection: split each triangle's uniquely labeled reference edge at its midpoint and use the inherited labels for its children, as defined in Compatible 2D newest-vertex-bisection meshes. Each closure call uses the exact finite-array interface of Dorfler marking and conforming newest-vertex refinement. The budget-doubling loop is safe because budget-exhausted returns the original mesh, and recursive closure terminates for every finite request by Termination and cumulative complexity of NVB conformity closure.