Matching simplex subdivisions give a NURBS knot-cell geometry interface

In the normalized cube every point has some ordering of its $d$ coordinates, hence belongs to at least one displayed simplex; strict order specifies exactly one interior. For a permutation $\pi$, the closed ordered region is the convex hull of the chain vertices $0,e_{\pi (d)},e_{\pi (d)}+e_{\pi (d-1)},\dots ,{\sum }_{r=1}^d{e}_{\pi (r)}$: the barycentric coefficients are $1-t_{\pi (d)}$, $t_{\pi (d)}-t_{\pi (d-1)},\dots ,t_{\pi (2)}-t_{\pi (1)},t_{\pi (1)}$. Thus a tie between consecutive coordinates sets exactly the corresponding barycentric coefficient to zero. For two permutations, the common weak order groups precisely the coordinates whose order is reversed into equal blocks; their intersection is the convex hull of the common chain vertices at the block boundaries, hence is a whole face of each simplex. This proves the subdivision is a simplicial complex with disjoint interiors. On a rectangular face one coordinate is $0$ or $1$; the remaining subdivision orders the remaining normalized coordinates. Adjacent rectangles have the same intervals in those coordinates and the same increasing-axis convention, so their induced face subdivisions coincide. The affine diagonal maps preserve these face identities. Therefore the finite subdivision is a matching affine simplicial mesh as required in [F2].

On a closed rectangle each map is a quotient of finite polynomials with denominator bounded below by a positive number. Its cell expression therefore extends smoothly across the rectangle into a neighborhood. Its derivatives, including second derivatives, are bounded there after shrinking the neighborhood. The determinant extends continuously with a positive lower bound on the closed cell. Its restriction to each closed simplex is injective by global injectivity of the map on $\bar{Q}$. This restriction actually extends injectively to some neighborhood of that simplex: otherwise there exist distinct pairs of points at distance at most $1\mathrm{/}n$ from the simplex with equal images. Compactness [F3] gives convergent subsequences with limits in the simplex; equality of images and injectivity force the same limit, contradicting local injectivity [F3] near that limit. A smaller neighborhood retains positive determinant, so the restriction is a neighborhood diffeomorphism. These arguments apply separately to $F$ and $G$.

Continuity of the original maps makes the restricted traces agree on shared simplex facets, including those lying on original knot planes. Global injectivity preserves disjoint interiors and makes intersections of physical cells precisely the images of the corresponding reference intersections. Their union is the original image. The map, inverse, determinant and finite second-derivative bounds from step 1.2 give all curved-cell requirements [F2]. The reference simplices here are actual parameter simplices; if one instead precomposes with a map from a unit simplex, its Jacobian scaling factors must be retained.

The cell restrictions do not alter the global maps, their physical images, or their value and almost-everywhere derivative differences. Every introduced reference-simplex facet lies in an affine hyperplane, the image of an affine $C^1$ map from ${\mathbb{R}}^{d-1}$ into ${\mathbb{R}}^d$. By [F4] and [A1] each such hyperplane is null, and the finite union of introduced facets is null. Thus taking essential suprema over the refined interiors gives the same $e_1$; continuity gives the same $e_0$. To state separate finite error bounds as required by [F2], write $P_{\alpha }^F,P_{\beta }^G$ for the two finite control-point lists and $L_F,L_G$ for their admissibility derivative bounds. The nonnegative rational coefficients in [F1] sum to one, so $e_0\le {\max }_{\alpha }\mathrm{\mid }{P}_{\alpha }^F\mathrm{\mid }+{\max }_{\beta }\mathrm{\mid }{P}_{\beta }^G\mathrm{\mid }$; the operator-norm triangle inequality gives $e_1\le L_F+L_G$. The two positive determinant lower bounds are their respective admissibility constants $j_{-}^F,j_{-}^G$. These coarse bounds establish the finite geometry interface; no smallness or mesh-dependent convergence follows from admissibility alone. Since both maps are bi-Lipschitz bijections of the closed box, $G\circ F^{-1}$ is the required homeomorphism between the closed image domains. The within-mesh trace matching in step 2.1 supplies boundary-trace compatibility in [F2]; no exterior trace equality between distinct maps has been proved.

Each normalized ordered-coordinate simplex belongs to a finite set depending only on $d$. Under diagonal scaling, diameter increases by at most ${\max }_r{\mathrm{\ell }}_r$ and inradius decreases by at most the factor ${\min }_r{\mathrm{\ell }}_r$ relative to the normalized simplex. Hence the diameter/inradius ratio is at most $\kappa$ times the largest ratio in that fixed finite set. This proves the stated reference shape bound. The remaining uniform curved-map bounds are exactly the additional displayed family assumptions; finite-grid smoothness in step 1.2 supplies only finite-grid constants. No analysis-space change was made in the subdivision.