Proof. Extend \(e(x)=\min\{a(x),1,L\mathop{\mathrm{dist}}(x,\mathbb R^n\setminus U)\}\) by zero outside \(U\), and put \[d(x)=\inf_{y\in\mathbb R^n}\{e(y)+L|x-y|\}.\] This function is \(L\)-Lipschitz and is bounded above by \(e\). It is positive at an interior point \(x\): on a small closed ball about \(x\), \(e\) has a positive minimum, and outside that ball the distance term has a positive lower bound.

For a smooth minorant take a smooth locally finite partition of unity \(\{\chi_i\}\) with compact supports in \(U\) and put \(b=\sum_i c_i\chi_i\), with positive constants \(c_i\). Given positive continuous bounds \(v,w\), choose \[c_i\le 2^{-i-2}\min\left\{ \inf_{\mathop{\mathrm{supp}}\chi_i}v, \frac{\inf_{\mathop{\mathrm{supp}}\chi_i}w}{1+\|D\chi_i\|_\infty} \right\}.\] Then \(b>0\), \(b\le v\), and \(|Db|\le w\). Replace a finite list of requirements by their minimum. For countably many regional tests use a locally finite compact cover with compact enlargements, refine the bounds on each intersecting support, and allocate total errors by a convergent positive series. Only finitely many regional requirements meet a given compact support. ◻

Proof. Choose a closed neighborhood \(K^+\) of \(K\) contained in the open set on which \(f\) is PL. Transport the PL structure of \(N\) to \(M\) by \(f\). The identity from the original structure to the transported structure is PL near \(K^+\). Hamilton’s relative approximation Theorem (Hamilton 1976, sec. 3, Theorem 2.2, pp. 68–69) applies to these two structures. Use the compatible metric \(d_f(x,y)=d_N(f(x),f(y))\) and the fine tolerance \(\eta\). It gives a homeomorphism \(a\) that is PL between the two structures, is the identity on \(K^+\), and satisfies \(d_f(a(x),x)<\eta(x)\). Then \(f_1=f\circ a\) has all the asserted properties. The boundary condition in Hamilton’s Theorem is precisely the additional condition stated here; for an open Euclidean domain its manifold boundary is empty. ◻

Proof. Put \(E=C\cup D\cup\partial B\). The deformation Theorem of Edwards–Kirby (Edwards and Kirby 1971, Theorem 5.1), applied at the inclusion of \(O\) in \(\mathbb R^3\), deforms every sufficiently close embedding \(e\) through embeddings \(e_t\), starting at \(e_0=e\), so that \(e_1=\operatorname{id}\) on \(E\). The deformation is unchanged outside a fixed compact neighborhood \(L\) of \(E\) in \(O\), and fixes the inclusion. Its continuity at the inclusion permits us to require \[|e_t(x)-x|<\lambda/2\quad(x\in L,\ 0\le t\le1), \qquad |e(x)-x|<\lambda/2\quad(x\in L).\] A sufficiently small uniform \(\delta\) on \(O\) meets the finitely many compact-open conditions that specify this neighborhood of the inclusion. In the terminology of that Theorem a proper embedding respects manifold boundaries; this condition is automatic for the ambient manifold \(\mathbb R^3\).

All the open images \(e_t(O)\) coincide. To see this, enclose \(L\) in a finite union of relatively compact subdomains of \(O\) whose boundary collars are outside the region where the deformation changes the map. The embeddings agree on that collar. Invariance of domain (Hatcher 2002, Theorem 2B.3) and degree relative to the common boundary show that the images of the enclosed subdomain are the same. Outside it the maps already agree. On this common open image define \[A_0=e\circ e_1^{-1},\] and define \(A_0\) to be the identity elsewhere. The map is already the identity off the compact set \(e_1(L)\) in the common image, so this is an ambient homeomorphism. It agrees with \(e\) on \(E\) and has displacement less than \(\lambda\). Since it fixes \(\partial B\) pointwise, it preserves the bounded complementary component \(\operatorname{int}B\). Restrict \(A_0\) to \(B\) and extend it by the identity outside \(B\) to obtain \(A\).

The construction uses only the displayed source configuration and continuity at the inclusion. For finitely many normalized configurations take the minimum of the resulting positive tolerances. ◻

Proof. Write \(d(y)=\mathop{\mathrm{dist}}(y,\mathbb R^3\setminus\Omega')\) and \(g_t=(1-t)h+tg\). Condition (5) puts the entire segment \(g_t(x)\) in \(\Omega'\). Since \(d\) is 1-Lipschitz, \[d(g_t(x))\le\frac32d(h(x)).\] If \(K\Subset\Omega'\) and \(m=\min_Kd>0\), the inverse image of \(K\) under the homotopy lies in \[h^{-1}\bigl(\{y\in\Omega':d(y)\ge2m/3\}\bigr)\times[0,1].\] The set in braces is compact because \(\Omega'\) is bounded. Thus \(g_t\) is a proper homotopy. Its endpoint \(g\) has the same degree as \(h\), namely one. A proper local diffeomorphism has finitely many preimages of each point, and here each preimage has positive local degree. The degree-one identity therefore says that every point of \(\Omega'\) has exactly one preimage. The local smooth inverses patch to a smooth inverse on \(\Omega'\). ◻

Proof. An orientation-reversing Euclidean isometry in the target reduces the proof to the orientation-preserving case. Fix a locally finite affine triangulation for \(h\). We prescribe summable error budgets for its edges, vertices, and a locally finite family of compact sets covering the remainder of the domain. All modifications below can also obey an arbitrary positive continuous value tolerance.

Proof. Choose a positive size function \(s\) on \(\Omega\) with sufficiently small absolute Lipschitz constant, below all the prescribed local bounds and below \(10^{-3}\mathop{\mathrm{dist}}(x,\mathbb R^3\setminus\Omega)\). Lemma 2 supplies such a minorant. Take the maximal dyadic cubes for which \[l_Q\le\inf_Qs.\] Failure of the condition for the dyadic parent gives \(s(x)\le(2+C\mathop{\mathrm{Lip}}(s))l_Q\) on \(Q\). The small Lipschitz constant then gives local comparability on \(B_Q(50)\). For example, choosing it sufficiently small makes every dyadic ratio in this neighborhood belong to \(\{1/2,1,2\}\). The distance bound puts \(B_Q(100)\) inside \(\Omega\). Maximality gives a decomposition, and the positive lower bound for \(s\) near every compact subset gives local finiteness.

The cube vertices have dyadic coordinates. Bounded local scale ratios and bounded normalized distance therefore give only finitely many normalized configurations. To triangulate, decompose each cube face into its square contact facets with adjacent cubes. Put every contact vertex and hanging subdivision point on the perimeter of each such square. Cone these perimeter segments from the square center, and then cone the resulting triangulation of each cube boundary from the cube center. The constructions on shared facets agree. They give nondegenerate simplices, and their normalized shapes range over a finite list.

Local comparability bounds the degree of the graph joining cubes whose \(B_Q(6)\) boxes meet. A coloring with one more than that degree proves (iii). Finally choose the initial local upper bounds so small relative to the coarse Whitney sizes that the finitely many required patch layers stay in fixed coarse neighborhoods. Since the coarse enlarged cover is locally finite, its interaction graph is locally finite. Taking a still smaller minorant \(s\) does not change any of these conclusions. ◻

Proof. Scale the patch to \(l_Q=1\). First reduce \(a\) using \(K_P\) so that \(h_Q(B_Q(3))\subset G(\operatorname{int}B_Q(4))\). Indeed, for \(x\in B_Q(3)\) and \(z\in\partial B_Q(4)\), the distance \(|H(z)-H(x)|\) is bounded below by \(K_P^{-1}\). The small errors in Equation (15) preserve this separation. Degree on \(\operatorname{int}B_Q(4)\), comparing \(G\) with \(H\) on the boundary, shows that \(h_Q(x)\) is in its image. The same statement holds with a small margin around \(B_Q(3)\).

The embedding \[\eta=G^{-1}\circ h_Q\] is consequently defined there. If \(z=\eta(x)\), then \[ |z-x|\le K_P|H(z)-H(x)| \le K_P\bigl(|H(z)-G(z)|+|h_Q(x)-H(x)|\bigr) <2K_Pa. \tag{16}\] Thus its closeness to the inclusion uses no inverse-Lipschitz bound for \(G\).

Take a fixed closed neighborhood \(C\) of \(B_Q(2)\) and fixed open neighborhoods \(O_0,U_0\) with \(C\subset O_0\Subset U_0\Subset\operatorname{int}B_Q(3)\), and let \(D\) be the union of the fully-shrunk older boxes intersected with \(B_Q(5)\). The half/full shrink margins give neighborhoods on which the map \(\eta\) on \(U_0\) agrees with the identity near \(D\). Include also an identity neighborhood of \(\partial B_Q(5)\), separated from \(C\). Choose these neighborhoods once for each normalized source configuration. By Equation (16), sufficiently small \(a\) makes the union of \(\eta\) and these identity maps an embedding, as in Equation (4). Apply Lemma 5 with \(B=B_Q(5)\) and displacement \(\lambda\). Precomposition by the resulting \(A\) installs \(h_Q\) on \(C\) and preserves the fully-shrunk boxes. Rescaling proves the Lemma. ◻

Proof. Fix the coarse neighborhoods, local bounds \(K_P\), and finite mesh configurations of Lemma 9. Until the final resolution choice, we describe rules and libraries uniformly over all admissible meshes, rather than fix a particular mesh. Write \(S=2n_0\) for the number of stages. A newly installed patch has protected radius 2. At each subsequent stage decrease the radius of every older protected patch by \[ d_*=(4n_0+4)^{-1} \tag{18}\] in its own \(B_Q\) coordinates. Its final radius is greater than \(3/2\), and in particular its closed original cube \(Q\) remains in the interior of its protected core. At an insertion we use half of the current prescribed decrease for agreement of the new model and the old map, and the full decrease for the sets fixed by the ambient change. These are precisely the margins in Lemma 10.

Choose all insertion and displacement tolerances by the backward procedure following Equation (17). They can be chosen to give an arbitrarily small final normalized value error on every patch. If \(a_{s,P}\) is the accuracy required of a new model at stage \(s\), require the current \(G-H\) bound before that stage to be less than \(a_{s,P}/8\), also on its interacting neighborhoods. This additional fraction is imposed during the same backward recursion. All these numbers are now fixed.

Proof of Theorem 3. Apply Proposition 11 with Sobolev tolerance \(\varepsilon/2\) and value tolerance \(\xi/2\), obtaining a PL homeomorphism \(h:\Omega\to\Omega'\). Apply Proposition 7 to \(h\) with the same tolerances. The triangle inequality proves Equation (2). Both approximations have exactly the target \(\Omega'\). Choosing \(\varepsilon\downarrow0\) gives the sequence asserted by the smoothing reduction. ◻

Proof. At a point belonging to a closed piece, only pieces containing that point can meet every sufficiently small neighborhood of it. At almost every point of an overlap, that point is a density point of the overlap in its parameter manifold. Two \(C^1\) extensions whose values agree there have equal derivatives in its tangent directions: their difference vanishes on a set of density one, and its first-order expansion therefore vanishes on every tangent vector. Continuity of the finitely many extension derivatives now gives the asserted essential limits. Apply this argument on the boundary parameter manifold for boundary directions. For a composition, partition by the finitely many choices of formulas and use the chain rule; tangential agreement on a preimage overlap follows from the same density argument. In a cone extension, away from its center and face rays the formula is a smooth radial factor times the boundary formula. The asserted limits consequently follow from those of the boundary derivative; the excluded rays and parameter exceptions have measure zero. ◻

Proof. We first majorize the pullback metric by a smooth metric that is conformally Euclidean on an added collar. Conformal coordinates on a rectangle then give the weak energy estimate. For the sharp estimate, coordinate tests force both the rectangle and its boundary correction to be nearly isometric.

Proof. Rescale to \(\ell=1\) and place an incenter at the origin. Cone-extend \(\sigma\) from that center; this is bi-Lipschitz for the fixed data. In radial coordinates \(x=rz\), \(z\in\partial D\) and \(0\leq r\leq1\), precompose the resulting map by the radial homeomorphism with profile \[q_{\alpha,\delta}(r)= \begin{cases} (1-\delta)r/\alpha,&0\leq r\leq\alpha,\\ 1-\delta+\delta(r-\alpha)/(1-\alpha),&\alpha\leq r\leq1. \end{cases}\] Here \(0<\alpha,\delta<1\), and the resulting parametrization is \[F_{\alpha,\delta}(rz) =H\bigl(q_{\alpha,\delta}(r)\sigma(z)\bigr).\] On \(r\leq\alpha\) all derivatives are at most a fixed-data constant times \(\alpha^{-1}\); hence the energy there is \(O(\alpha^{m-p})\). On \(r>\alpha\) the radial derivative tends to zero as \(\delta\downarrow0\), while the tangential derivatives tend essentially to \(r^{-1}D_t(H\circ\sigma)(z)\). Indeed, the angular derivative contains \(q_{\alpha,\delta}(r)DH(q_{\alpha,\delta}(r)\sigma(z))D_t\sigma(z)\). The last factor is evaluated at the fixed boundary point \(z\); only the base map’s tangential derivatives require a limiting argument. The radial Jacobian is comparable to \(r^{m-1}\) with constants determined by the cell shape. Essential compatibility and dominated convergence yield an upper bound \[C\left(\int_\alpha^1r^{m-1-p}\,dr\right) \int_{\partial D}|D_t(H\circ\sigma)|^p\] for the limiting outer energy. To apply the essential limits, use a sequence of the individual bi-Lipschitz radial changes and the common full-measure set of their Lebesgue representatives. Coning \(\sigma\) only inserts its fixed a.e. angular derivative. Since \(p<m\), the radial integral stays bounded as \(\alpha\downarrow0\) and the central error tends to zero. First choose \(\alpha\) and then \(\delta\). Restoring scale multiplies the boundary integral by \(\ell\) and proves (36). All radial profiles fix the boundary, which proves the gluing assertion. ◻

Proof. Choose a small topological ball \(B\) about \(z\) in which \(P\) is a proper unknotted disk. All changes will occur away from \(\partial B\). Straighten \(I\) in its supplied coordinates near \(z\) and choose two cap levels on opposite sides of \(z\) within a segment of the axis whose only intersection with \(P\) is \(z\). In a parameter disk for \(P\) choose round disks \(D_1\Subset\operatorname{int}D_0\) about the parameter of \(z\) so that \(P(D_0)\) is compactly contained in the open slab between these levels and in the straight-axis chart. Now choose a sufficiently thin polygonal cylinder \(C\) about the intervening axis interval. Its caps miss \(P\). Compactness and the isolated-axis intersection make all intersections with its side annulus \(A\) lie in \(P(D_1)\) once the radius is sufficiently small. Every side loop and its bounded parameter subdisk then lie in \(D_1\). Thus all disks subsequently used for linking lie inside the cap slab and cannot meet the axis extended beyond the isolated supplied segment.

Put \(P\) in PL general position near \(A\), without changing it near the axis or near \(\partial B\). To justify this preliminary operation, write \(P\) as the image of a flat disk under a chart homeomorphism, approximate that homeomorphism by a PL homeomorphism on a compact neighborhood of the prospective side intersections, and install the approximation on a smaller closed neighborhood by Lemma 5. The support misses the axis, and the unsupported part of the disk has a positive gap from the cutting side. Take the perturbation sufficiently small that \(P(D_0)\) remains in the open cap slab and all prospective side intersections remain in its parameter interior. A small change of the polygonal side gives transverse intersections. Consequently \(P\cap A\) is a finite family of polygonal simple closed curves.

First remove every curve which is null-homotopic in \(A\). Choose an innermost such curve on \(A\). Its annular disk has interior disjoint from \(P\): it contains no further null curve by the choice, and cannot contain an essential curve of \(A\). Replace the disk which this curve bounds in \(P\) by that disk on \(A\), pushed slightly off \(A\). Round the seam on the side dictated by the remaining part of \(P\). The annular disk lies inside the open cap slab; its small push can be kept there. The operation replaces a parameter subdisk of \(D_0\), and hence the remaining relevant subdisks still lie in that slab. It gives an embedded locally flat disk, removes the chosen intersection curve, and creates no new side or axis intersections. Repeating finitely many times leaves only essential side curves. The disks discarded in this operation may contain \(z\); in that event the axis intersection is simply removed.

Every remaining essential side curve has linking number \(\pm1\) with the extended straight axis. Its disk in \(P\) must therefore meet that axis. Since the surgeries have added no axis intersections, this disk contains the original isolated intersection. The disks bounded by the remaining curves are thus nested. Replace the outermost one by a tame proper disk inside \(C\) with the same rim and with exactly one flat transverse crossing of the axis. Such a disk is obtained by isotoping the essential rim in the side annulus to a circular section and extending that isotopy across an outer collar of a meridian disk. After this replacement the rest of \(P\) is outside the cylinder. If no side curves remain, there is no intersection with the cylinder at all: the caps miss \(P\), the outer boundary of the proper disk lies outside \(C\), and any component entering \(C\) would have to cross its side.

The result is a locally flat proper disk in \(B\) agreeing with the original disk near its boundary. Both disks split \(B\) into balls. The locally flat Schoenflies theorem (Brown 1962, Theorem 4), followed by extension on the two sides, gives an ambient homeomorphism of \(B\), fixed on \(\partial B\), carrying the original disk to the constructed disk. Extend it by the identity outside \(B\). The ball, the cylinder, and each surgery support can be chosen successively smaller, proving the support assertion.

Finally, in the opposite-side case retain two nearby axis points on opposite sides, outside the modification support. Separation forces the modified disk to meet the intervening interval. Since the construction leaves at most one intersection, exactly one remains. ◻

Proof. After cleaning, use the indicated coordinates so that the relevant source and image surface germs are planes. Choose a small outer source ball in this germ and a much smaller inner source ball. Prescribe an affine map on the inner ball, sending its equator to the target plane, with its image contained in the outer image ball. Choose the affine map to agree with the side correspondence and orientation of \(h\).

Extend first across the planar annulus between the two equators. On either side, the outer half-ball is a topological ball with a flat boundary patch near the equatorial center. Removing the inner half-ball, attached along a disk in that patch, again leaves a ball. This follows either from a boundary half-space chart and relative Schoenflies or from the usual ball-with-boundary-attached-ball model. The resulting boundary homeomorphisms extend across these balls. The two extensions agree on the planar annulus, so they define the required cutoff of \(h\). The surface remains in the target plane during this further change. In the inverse construction the inverse affine germ is again affine. Finally, continuity of \(h\) and \(h^{-1}\) permits the paired support neighborhoods to be made small in both spaces. ◻

Proof. Choose a tube about \(K\) meeting \(\partial B\) in two meridian disks. Such tubes can be constructed on the two cut intervals, starting with the product charts at their endpoints and continuing ordinary regular neighborhoods along their interiors. Glue the endpoint disks to obtain the tube about \(K\). The exterior of the whole tube has fundamental group \(\mathbb Z\), generated by a meridian. Its two pieces are glued along the annulus obtained from \(\partial B\) by deleting the meridian disks. The inclusion of this annulus into the whole exterior induces an isomorphism on fundamental groups. Hence its inclusions into the two pieces are injective. The injective-amalgam form of van Kampen then embeds each piece’s group in the total group. Since its image contains the meridian generator, each piece has group \(\mathbb Z\), again generated by that meridian.

In the exterior piece lying in \(B\), join the two end circles by an arc on the annulus from \(\partial B\), and close it by a connector on the lateral annulus of the drilled tube. This is a simple loop on the boundary of the exterior. Its class is a power of the meridian. Twisting the connector by the opposite integer power on its annulus makes the loop null-homotopic while preserving simplicity. Dehn’s Lemma supplies a properly embedded disk with that boundary.

Restore the tube and attach the radial strip joining its connector to the core arc. The strip is embedded even when the connector twists: in product coordinates its angle may vary with the interval coordinate while its radius decreases to zero at the core. Its two short edges lie in the meridian end disks. The resulting locally flat disk has boundary consisting of \(K\cap B\) and an arc in \(\partial B\). Thus the arc is boundary-parallel. Sliding along a neighborhood of this disk identifies the pair with a ball and a standard diameter. Rounding the seams does not change this conclusion. ◻

Proof. Write \(P=S^2\setminus\{v_1,\ldots,v_n\}\) and \(S^\circ=S\setminus\{z_1,\ldots,z_n\}\). Radial projection restricts to \[f:S^\circ\longrightarrow P.\] This map is proper: its extension to the compact sphere \(S\) has \(f^{-1}(v_i)=\{z_i\}\), so the inverse image of a compact subset of \(P\) is a compact subset of \(S^\circ\). Orient \(S\) as the boundary of its bounded complementary region. Since \(S\) encloses the inner sphere, its unpunctured radial projection has degree \(1\). Local degree at any point of \(P\) therefore shows that the proper map \(f\) also has degree \(1\).

We record explicitly the covering argument for \(f_*\). Let \(p:\widetilde P\to P\) be the connected covering corresponding to \(f_*\pi_1(S^\circ)\), and let \(\widetilde f\) be the lift of \(f\). The lift is proper: for compact \(K\subset\widetilde P\), its inverse image is closed in the compact set \(f^{-1}(p(K))\). Give the covering its induced orientation and let \(d\) be the proper degree of \(\widetilde f\). For a fixed \(y\in P\), compactness of \(f^{-1}(y)\) implies that its lifted image meets only finitely many points of the discrete closed fibre \(p^{-1}(y)\). Additivity of local degree expresses \(\deg f\) as the sum of the local degrees over these points. At every point of the covering the local-degree sum for \(\widetilde f\) equals \(d\); it is zero at a point outside its image. Consequently an infinite-sheeted covering would force \(d=0\) and then \(\deg f=0\). For a finite-sheeted covering with \(k\) sheets, the same calculation gives \(1=k d\). Hence \(k=1\), and \(f_*\) is surjective. Both fundamental groups are free of rank \(n-1\), so the Hopf property of finitely generated free groups (Harpe 2000, III.A, Complements 18–19) makes \(f_*\) an isomorphism.

Choose pairwise disjoint closed regular-neighborhood tubes \(N_i\) around the full strands \(L_i\), each meeting \(S\) in one meridian disk and meeting each boundary sphere of \(M\) in one end disk. Such tubes are obtained from the pair charts at the crossings and regular neighborhoods of the two remaining tame subintervals, using the same end disks where they join. In particular all tubes and their pieces on either side of \(S\) are standard interval neighborhoods. Drill out their interiors relative to \(M\), obtaining an exterior \(E\). Denote its bottom and top planar boundary surfaces by \(F_0\) and \(F_2\), and the truncated copy of \(S\) by \(F_1\). Regular-neighborhood coordinates identify the fundamental groups of these exteriors with those obtained by deleting just the core strands, compatibly with the inclusions of the truncated surfaces. Since \[M\setminus\bigcup_iL_i\cong P\times[a,b],\] our calculation of \(f_*\) shows that \(F_1\hookrightarrow E\) induces a fundamental-group isomorphism. The same is immediate for \(F_0\) and \(F_2\).

Cut \(E\) along \(F_1\), and let \(W\) be the piece on the inner side. The inclusion of \(F_1\) into either piece is injective on fundamental groups, because its composite into \(E\) is injective. Van Kampen and the normal-form Theorem for a free product with amalgamation therefore embed both piece groups in \(\pi_1(E)\). As the common group \(\pi_1(F_1)\) already maps onto \(\pi_1(E)\), both piece inclusions are isomorphisms. It follows that both \[\pi_1(F_0)\longrightarrow\pi_1(W),\qquad \pi_1(F_1)\longrightarrow\pi_1(W)\] are isomorphisms. Apart from these horizontal surfaces, the boundary of \(W\) consists of \(n\) annuli \(A_i\), one from each drilled tube, joining the corresponding boundary circles of \(F_0\) and \(F_1\).

The manifold \(W\) is irreducible. Indeed, let \(\Sigma\) be a locally flat sphere in its interior. It cannot separate the boundary spheres of \(M\), since every full strand joins those spheres and is disjoint from \(\Sigma\). Schoenflies therefore gives a ball \(B\) bounded by \(\Sigma\) and contained in \(\operatorname{int}M\). Each \(N_i\) is connected, is disjoint from \(\Sigma\), and meets \(\partial M\), so it lies outside \(B\). The connected surface \(F_1\) also lies outside \(B\), since it is disjoint from \(\Sigma\) and its boundary meets the tubes. Thus \(B\subset W\). All subsequent disks and spheres may be taken locally flat, or PL after triangulating this compact \(3\)-manifold.

We now give a relative product recognition argument, including the boundary identifications. Orient \(F_0\) and \(F_1\) so that oriented peripheral loops correspond across the annuli \(A_i\); these are opposite to one another when compared with their induced orientations as parts of \(\partial W\). The fundamental-group identification between them preserves the individual oriented peripheral conjugacy classes. The surface classification Theorem in its peripheral-preserving Dehn–Nielsen form supplies a homeomorphism \(j:F_0\to F_1\) realizing this outer isomorphism and matching the labelled boundary circles (Farb and Margalit 2012, Theorem 8.8 and Proposition 3.19).

Here is the adjustment from outer to based identifications. For this step relabel the boundary circles and their annuli by \(0,\ldots,n-1\). Choose a basepoint \(x_0\) on one boundary circle \(C_0\) of \(F_0\), and a connector \(\Delta_0\) in its annulus from \(x_0\) to \(j(x_0)\). Transport the second inclusion into \(W\) along \(\Delta_0\). Its composition with \(j_*\) and the first inclusion differ by an inner automorphism. Both carry the positive based peripheral of \(C_0\) to the same element \(g_0\) of \(\pi_1(W,x_0)\), so the conjugating element centralizes \(g_0\). Every peripheral of an \(n\)-holed sphere is primitive in its free fundamental group; since \(n\geq3\), its centralizer is precisely the cyclic group it generates. Twisting \(\Delta_0\) by an appropriate integer power in its annulus therefore gives the based equality \[[c]=[\Delta_0\,j(c)\,\Delta_0^{-1}] \quad\text{in }\pi_1(W,x_0) \quad\text{for every based loop }c\text{ in }F_0. \tag{*}\]

Choose disjoint properly embedded arcs \(\alpha_1,\ldots,\alpha_{n-1}\) in \(F_0\), joining \(C_0\) to the other boundary circles, whose cutting opens \(F_0\) to a disk. Their endpoints on \(C_0\) are distinct. For each other circle choose the endpoint \(x_i\) of its arc, a path \(c_i\) from \(x_0\) to \(x_i\) following \(C_0\) and then \(\alpha_i\), and an initial connector \(\Delta_i\) in \(A_i\) from \(x_i\) to \(j(x_i)\). The two paths \[c_i\Delta_i\quad\text{and}\quad\Delta_0j(c_i)\] transport the peripheral at \(j(x_i)\) to the same based element of \(\pi_1(W,x_0)\), by (*) and the annulus correspondence. Their discrepancy thus centralizes that peripheral. The same centralizer argument allows an integer twist of \(\Delta_i\) making these two paths homotopic relative to their endpoints in \(W\).

Parametrize each \(A_i\) as a circle product, with its lower boundary the identity, its upper boundary the restriction of \(j\), and its chosen connector in the adjusted relative homotopy class. Such a parametrization exists because its remaining integer winding is changed by a Dehn twist of the annulus. On the base annulus use the parametrization with track \(\Delta_0\) at \(x_0\) and use its tracks also at every endpoint of an \(\alpha_i\) on \(C_0\). Thus all connectors are disjoint. The product tracks are compatible with travel along a boundary circle, so the preceding path equalities imply that each closed curve formed from \(\alpha_i\), its two connector tracks, and \(j(\alpha_i)\) is nullhomotopic in \(W\). These curves, denoted \(\gamma_i\), are disjoint simple curves on \(\partial W\).

Dehn’s Lemma supplies properly embedded disks \(D_i\subset W\) with \(\partial D_i=\gamma_i\) (Papakyriakopoulos 1957). They can be chosen disjoint: put them in general position and remove their intersection circles by the usual innermost-disk replacements, keeping their already disjoint boundaries fixed. The chosen horizontal and annular maps give a boundary homeomorphism from \(\partial(F_0\times[0,1])\) to \(\partial W\) carrying the boundaries of the rectangles \(\alpha_i\times[0,1]\) to the \(\gamma_i\). Consequently cutting \(W\) along the \(D_i\) produces a manifold whose boundary is one sphere, just as cutting \(F_0\times[0,1]\) along those rectangles does. Every component of a manifold obtained by these proper disk cuts has nonempty boundary, so the cut manifold is connected. Irreducibility persists under the cuts: a sphere in the cut manifold bounds a ball in \(W\), and each cutting disk, being connected and having boundary on \(\partial W\), must lie outside that ball. A collar inset of the spherical boundary then shows that the cut manifold is a ball.

Extend the boundary correspondence over the rectangles and the \(D_i\), and then over the resulting balls. Regluing gives a product homeomorphism \(F_0\times[0,1]\to W\) realizing the chosen boundary correspondences. Restore each removed tube piece. Extend its boundary-circle maps over its two cap disks, taking the marked core endpoints to one another. A homeomorphism of the boundary of a standard ball–interval pair preserving the two endpoints extends to the pair: identify both pairs with a ball and a diameter and cone the boundary map. Applying this to each tube extends the exterior product to the asserted product of the filled region and its labelled strands. At the lower end choose the cap extensions to agree with the identity, so that \(\Phi(q,0)=a q\).

Finally the formula \[(q,t)\longmapsto\frac{\Phi(q,t)}{|\Phi(q,t)|}\] is the claimed boundary-comparison homotopy. It fixes each \(v_i\) because its product track is contained in \(L_i\), and it avoids all marked directions whenever \(q\) is unmarked because the product homeomorphism is a homeomorphism of pairs. ◻

Proof. We first straighten the distorted arc systems within their separate angular sectors. Each distorted arc is tame as an ambient arc, being the image or preimage of a straight interval under a homeomorphism. At each of its intersections with a chosen round section, straighten the arc in an ambient chart and apply Lemma 16 to the section disk. The opposite-side crossing persists once if the support is sufficiently small. Move the changed section back to the round section by the inverse supported homeomorphism, dragging the arc with it. The resulting arc has a standard transverse crossing with the original round section. Choose these changes in disjoint annular patches inside the relevant angular sectors. They can be made arbitrarily small and preserve nesting of the sections. All section crossings are now standard pair crossings.

Consider the connecting strand between two successive sections inside one angular sector. Their annular-sector region is a ball, and this strand is a proper interval in it. It is unknotted for the following reason. Choose a second arc of the same distorted system, which is possible by the cardinality hypothesis. On the undistorted side of the system, the two radial arcs can be closed outside a smaller germ neighborhood by a polygonal path to give an unknot. Take the closing polygon and its spanning disk in a sufficiently small original chart ball whose image remains in the working chart. The image of the closed curve is still an ambient unknot: the chart-ball homeomorphism extends across its exterior by locally flat Schoenflies. For a reverse system use \(h^{-1}\) in this argument. Choose the round sections so close to the center that they avoid the closing path. The second arc lies in its own disjoint sector, so the first sector’s annular ball meets this unknot in just the strand under consideration, with exactly two boundary crossings. The section-cleaning homeomorphisms preserve the ambient unknot. Lemma 18 therefore makes the strand boundary-parallel in its annular-sector ball.

Straighten each such strand in that ball, using matching maps on the cap disks and the identity on the sector sides. The prescribed cap maps present no extra obstruction: for the standard ball–diameter pair, a homeomorphism of the boundary sphere respecting the two ends extends by coning, and fixes the diameter as a set. At the outermost section make a transition inside the same sector and extend by the identity outside a slightly larger neighborhood. The infinitely many annular straightenings glue to a homeomorphism at the center, with continuous inverse there, because their supports lie in balls with radii tending to zero.

Perform the target changes for the forward system and the source changes for the reverse system. All the sectors are disjoint, so the target changes leave the target reverse rays fixed, and the source changes leave the source forward rays fixed. Pre- and post-composing accordingly gives a homeomorphism \(H\) carrying every labelled ray of the combined system to itself as a set, as a germ. The generator test is unchanged: its source loops and their image directions have positive margins from the supports. In the equatorial case the target changes have arbitrarily small angular displacement, since they act in sufficiently narrow sectors; thus the parametric angular test persists within its homotopy tolerance.

Choose a test radius \(r\) inside this exact-ray germ. The locally flat sphere \(S=H(\partial B_r)\) meets each marked ray once in a standard topological crossing. Indeed \(H\) carries the standard source sphere–ray pair to this pair. A sufficiently small round target sphere lies inside \(H(B_r)\); together with \(S\) it bounds a shell to which Lemma 19 applies. Its product comparison of the marked boundary spheres is homotopic to radial projection on their punctured complements.

In the first angular case the comparison of the outside boundary map with the intended identity map induces the identity on the chosen based generators. The punctured-sphere mapping-class Theorem implies that it is isotopic to the identity through maps fixing the labelled marked points (Farb and Margalit 2012, Theorem 8.8). The same Theorem detects the orientation here; the assumption \(n\geq3\) excludes the two-puncture ambiguity.

In the equatorial case take a consecutive spanning chain of equatorial joining arcs between marked points, omitting one closing arc. Their projected images represent the same classes relative to their marked endpoints. To see this directly, the middle of each parametrized image lies in a narrow corridor containing no puncture, while each short end lies in a disk containing only its own marked point. In such an end disk, possible angular winding can be interpolated using lifted polar angles on the open end of the arc; the radius tends to zero at the endpoint throughout the interpolation. Thus no framing obstruction remains at that marked point. Interior points of the image arcs avoid all marked points because \(H\) already maps the complete labelled ray germs exactly to themselves. The homotopy comparison in Lemma 19 transfers these arc classes to the boundary identification. The usual disjoint-arc isotopy test fixes these classes simultaneously (Farb and Margalit 2012, sec. 1.2.7, Proposition 2.8, and Lemma 2.9). Cutting along this chain leaves a disk, on which the orientation-preserving boundary identification is isotopic to the identity. This proves the required trivial marked mapping class also in the second case.

Choose a smaller source sphere \(\partial B_s\) whose intended identity image is contained in \(H(B_r)\). Product coordinates on the source annulus and on the target annulus, together with the preceding marked-sphere isotopy, extend the outside values of \(H\) and the inside identity across the annulus while respecting every labelled ray. Set the map equal to the identity on \(B_s\) and equal to \(H\) outside \(B_r\). This is a homeomorphism and supplies the asserted identity germ. The earlier sector straightenings and the final cutoff have arbitrarily small support in the corresponding spaces. Their angular bounds and the given strict margins prove the remaining assertions. ◻

Proof. First realize the required coordinate changes by supported \(C^1\) diffeomorphisms. After separating an affine first jet, such a local coordinate change has the form \(g(0)=0\), \(Dg(0)=I\). On a sufficiently small ball, a cutoff \[\widetilde g(x)=x+\chi(x)\bigl(g(x)-x\bigr)\] equals \(g\) on a smaller ball and the identity outside the larger one. Because \(Dg-I=o(1)\) and \(g(x)-x=o(|x|)\) at the center, its derivative is uniformly close to \(I\) when the balls are sufficiently small. It is therefore a supported \(C^1\) diffeomorphism. Use disjoint locally finite support neighborhoods in the source and target; the affine parts are absorbed into the prescribed affine germs. After these coordinate changes the homeomorphism is literally affine on neighborhoods of all protected points.

Enclose smaller germs by closed polyhedra inside those affine neighborhoods and apply the relative PL approximation of Lemma 4. Equivalently, excise these neighborhoods and approximate on their open complement as a PL three-manifold with boundary, relative to its already prescribed boundary collar. Only finitely many protected pieces occur on any compact, so the construction is locally finite. Undoing the supported \(C^1\) coordinate changes gives a locally bi-Lipschitz homeomorphism with the stated finite piecewise-\(C^1\) description on compacts.

The exact germs preserve the crossings there. Elsewhere the specified avoidances have positive gaps on the compact pieces where they are needed, so sufficiently fine approximation creates no additional intersections. The same choice preserves angular inequalities with a strict margin. Finally, for a homeomorphism fine source-dependent approximation tolerances can be chosen to ensure prescribed inverse tolerances on target compacts; use continuity of the original inverse and compact exhaustion. ◻

Proof. For \(m=1\), Sobolev slicing gives absolute continuity on almost every parallel line, and hence the one-dimensional area formula and Lusin property. A smooth curve chart can be foliated by transverse translates and straightened by a local ambient smooth diffeomorphism, giving the same conclusion in that chart. For \(m=2\) use (Csörnyei et al. 2010, Theorem 1.3): a \(W^{1,2}\) homeomorphism in dimension three has the two-dimensional Lusin property on almost every parallel plane. Together with Sobolev slicing and the usual Lipschitz decomposition of a Sobolev map (Bojarski and Hajłasz 1993, Theorem 3(1)), this gives the surface area formula. Apply the Csörnyei–Hencl–Malý theorem after straightening a smooth graph by an ambient smooth diffeomorphism. Fixing any other translation coordinates and then applying Fubini proves the assertion for the translated graph templates.

Stratify the semialgebraic source parametrizations \(P:K\to\Omega\), their source images \(P(K)\), and the ranks and overlaps of these parametrizations, using a common finite refinement as in (Hardt et al. 2011, Proposition 2.3 and Lemma 2.5). This stratification concerns \(P\), not the generally non-semialgebraic map \(f\circ P\). On a full-rank branch the preceding graph argument applies, and its parameter derivative is \(Df(P)DP\). On a lower-rank piece, use the slicing assertion for its lower-dimensional image stratum; for instance a one-dimensional image is rectifiable by the curve assertion and has zero two-dimensional measure. These observations also make images of parameter-null sets negligible on the regular branches. The chain rule and tangential integrability may alternatively be obtained by approximating \(f\) by smooth maps in ambient Sobolev norm and using Fubini to pass along a subsequence in parameter Sobolev norm.

We use the multiplicity area formula and the rectifiable-set coarea formula in (Simon 2014, chap. 2, Section 3, and Chapter 3, Section 2). The area formula and translation coarea imply that almost every complementary affine test avoids the exceptional image sets. At a remaining source value, stratification, local inversion on regular branches, and compactness leave a single local smooth source stratum with a fixed finite number of parameter branches. Dimension-drop intersections of different pieces are negligible by the same argument. More explicitly, full-rank strata of the finite refinement are diffeomorphic onto their images, which can be made identical or disjoint; their number, rather than compactness alone, bounds the branch multiplicity. On relatively compact branch charts \(P\) is bi-Lipschitz, so parameter-null sets first have null source image and then null target image by the separate sliced Lusin property. Finally, \(p\ge m\) in the admitted cases. Translation and Fubini give finite tangential \(m\)-mass on almost every compact template, and the area formula for its orthogonal projection makes the complementary intersection count finite for almost every affine test. The locally finite family involves only finitely many templates on each compact set; intersect their full-measure choices and then use a countable exhaustion. This proves all the simultaneous assertions. ◻

Proof. Choose a positive size function \(d(u)\) with sufficiently small absolute Lipschitz constant, satisfying the prescribed local upper bounds and \(d(u)\ll\mathop{\mathrm{dist}}(u,\partial V)\). Make it constant, with the desired common value, near each lattice patch. Outside deep lattice regions take a maximal packing at this scale, then independently jitter each retained site in a ball of a fixed small fraction of its local scale, with uniformly bounded density. In a lattice patch use a common uniform positional shift over one period. Truncate the unshifted lattice lists in their buffers, leaving overlap with the background cloud; the separated lattice patches remain many steps apart. The resulting sites cover at distance at most \(Cd(u)\) and have bounded counts within that distance. A minimum separation between every pair of sites is unnecessary.

Use their Delaunay subdivision, with a fixed pulling order to triangulate nonsimplicial cells. Its localization can be checked inside the open domain. An empty ball containing or passing through a point near \(u\) cannot have radius much larger than \(d(u)\): moving several mesh lengths from that point toward the center stays inside \(V\) and gives a location well inside the ball; the covering property then puts a site strictly inside it. The slow variation of \(d\) justifies using the same scale during this argument. Finite restrictions of the cloud therefore stabilize near \(u\) once they contain a sufficiently enlarged local cloud. Their ordinary Delaunay subdivisions cover and agree there. This proves local existence, local finiteness, and compatibility of the subdivisions on \(V\). Every candidate uses a bounded list of neighboring sites at comparable scales; coincidences occur with probability zero.

Deep in a rectangular lattice, the Voronoi description shows that the Delaunay cells are the rectangular boxes. Increasing lexicographic pulling gives their Kuhn simplices. Taking the scale small enough places any prescribed deep compact set and all its touching simplices within this region before sampling.

For the density assertion, first fix a candidate tuple of site indices, before imposing its Delaunay selection event. If it contains lattice sites, they all belong to one shifted lattice. Use its physical shift as \(t\) and write each unstructured site as \(x_0+t+dz_j\), with \(x_0\) a fixed candidate anchor. The original joint law of the shift and the \(k\) unstructured sites has density at most \(Cd^{-3(k+1)}\). Replacing the latter positions by \(z_j\) has Jacobian \(d^{3k}\), giving (37). If no lattice site is present, use one site as \(x_0+t\) and express the other \(k-1\) sites relative to it in units \(d\); the Jacobian \(d^{3(k-1)}\) gives the same bound from the original density \(Cd^{-3k}\). The normalized relative coordinates lie in a bounded set because all candidate sites have comparable scales and lie in one bounded-scale neighborhood. The Delaunay selection event is an indicator bounded by one and is discarded for an upper estimate. All subsequent calculations integrate the unnormalized translation slice for each \(z\), then integrate \(z\) over this bounded set. Thus no bound on a normalized conditional translation density is needed. ◻

Proof. Since the candidate \(\rho=\sigma\) gives \(b\leq b_\sigma\leq1\), \(a\geq0\). Every denominator \(\max\lambda_j\) is at least \(1/4\), so the normalized Lipschitz constant of \(b\) is absolute. The chain rule gives (40). On a face \(\tau\), replacing a candidate \(\rho\) by \(\rho\cap\tau\) leaves its nonzero outside coordinates unchanged and can only lower its constant, because \(b_{\rho\cap\tau}\leq b_\rho\). An empty intersection gives an outside-coordinate ratio of one and cannot improve on \(b_\tau\leq1\). Thus restrictions agree on faces. Positivity, zero coordinates, and the order of coordinates are preserved by (39).

We verify invertibility even though \(a\) is variable. Given output ratios \(z_i=T_i/T_{\max}\in[0,1]\), for a trial \(D\geq0\) let \(r_i\) be the unique solution of \[ z_i=r_i e^{-D(1-r_i)},\qquad 0\leq r_i\leq1. \tag{42}\] The function of \(r_i\) on the right is strictly increasing. Each \(r_i\) is nondecreasing in \(D\), with the zero and maximal ratios fixed at zero and one. Put \(R=\sum_i r_i\) and \(\lambda_i=r_i/R\). The trial value of \(a\) is \(DR\), an increasing function of \(D\) with increase at least that of \(D\). In (38) the outside-coordinate ratios are exactly \(r_i\); hence \(b\) at this trial inverse is nondecreasing in \(D\). Its prescribed value \(e^{1/b}-e\) is nonincreasing. The equation \[DR=e^{1/b(\lambda)}-e\] has a unique solution, by continuity, its sign at \(D=0\), and its positive sign for large \(D\). This proves bijectivity. On an individual closed simplex \(b\) has a positive lower bound, so the solution has bounded \(D\). Equation (42) has uniformly Lipschitz inverse in this bounded range, including at zero output coordinates. The scalar equation’s left-minus-right increase is at least the increase in \(D\), so its root depends Lipschitz-continuously on the output. Since \(T_{\max}\geq1/4\), forming the output ratios is Lipschitz as well. The forward formula is Lipschitz on that simplex, and its inverse is Lipschitz. The finite min/max alternatives in (38) give finitely many closed pieces with \(C^1\) extensions of the formulas. Local finiteness proves all the global assertions. If the incident constants are one, the minimum is one, so \(a=0\) and \(T=\operatorname{id}\).

It remains to prove the concentration assertion. On a compact trace set away from \((m+1)\)-fold leader ties, at most \(m\) coordinates can be near the maximum. Coordinates outside that set and all their derivatives are exponentially small in \(a\), multiplied only by powers of \(a\) and \(\log(a+e)\) by (40). The active coordinates have total mass one up to those errors and therefore take values in a face of dimension at most \(m-1\). Their \(m\)-Jacobian vanishes. Expanding the actual \(m\)-Jacobian then shows uniform convergence to zero away from the counted ties, including along simplex faces by the same piecewise calculation.

Near a transverse hit relabel the leaders \(0,\ldots,m\) and use \[u_i=\lambda_i-\lambda_0,\qquad 1\leq i\leq m,\] as trace coordinates. All leader coordinates have a fixed positive lower bound there, and the remaining coordinates are separated below them. Normalize the probabilities on the leaders first. Their logit differences are \[a(u)u_i+\log\frac{\lambda_i(u)}{\lambda_0(u)}.\] Inactive probabilities and derivatives are exponentially small. Write \(a_0=a(0)\). Since \(b(0)=1/\log(a_0+e)\) and \(b\) is uniformly Lipschitz, on \(u=t/a_0\) with bounded \(t\) one has \[\log\frac{a(u)+e}{a_0+e} =O\left(\frac{(\log(a_0+e))^2|t|}{a_0}\right)=o(1).\] The scaled derivative terms coming from \(Da\) tend to zero by (40). The leader map and its a.e. derivatives therefore converge locally uniformly, apart from its harmless piece seams, to \[t\longmapsto \left(\frac{e^{t_i}}{1+\sum_{j=1}^m e^{t_j}}\right)_{i=1}^m.\] This softmax map is a diffeomorphism from \(\mathbb R^m\) onto the interior of the standard \(m\)-simplex. Multiplication by the physical edge vectors consequently gives exactly the flat \(m\)-area of the leader face, provided the tails vanish.

Here are uniform tail bounds. The determinant of the active probability derivative is its probability product times the determinant of the logit derivative. The logit derivative has the form \(aI+u\otimes Da+B\), with \(B\) bounded for the fixed trace chart. The probability product is at most \(C e^{-ca|u|}\). Thus the absolute determinant, modulo exponentially small inactive terms, is bounded by \[ C(1+a+|Da|\,|u|)^m e^{-ca|u|}. \tag{43}\] For \(|u|\leq a_0^{-1/2}\), the preceding logarithmic estimate is uniform and gives \(a/a_0\to1\) there. After scaling, the determinant has an integrable bound \(Ce^{-c|t|}\), because \(|Da||u|/a_0\leq C(\log a_0)^2a_0^{-1/2}=o(1)\). For \[a_0^{-1/2}<|u|\leq c_1/\log(a_0+e),\] choose \(c_1\) small relative to the fixed Lipschitz bound for \(b\). Then \(a\geq a_0^{3/4}\) for large \(a_0\), and \(a|u|\geq a_0^{1/4}\). Exponential decay in (43) absorbs all polynomial and logarithmic factors uniformly for every larger \(a\). In the remaining sufficiently small neighborhood, \[b(u)\leq b(0)+C|u|\leq C'|u|, \qquad a(u)+e\geq e^{c'/|u|}.\] For small \(r=|u|\), the expression \(a^N e^{-car}\) is decreasing once \(a\geq e^{c'/r}/2\), for any fixed \(N\). Increasing \(N\) to absorb the logarithms in (43) therefore gives an integrable bound of the form \[C\exp(Nc'/r)\exp(-c''r e^{c'/r}),\] independent of how high the local floors are. At every fixed \(u\ne0\) the integrand tends to zero as the floor tends to infinity. Dominated convergence treats this last region. These three bounds prove tail vanishing and (41).

For a compact measured list there are only finitely many charts and hits. One may therefore prescribe the floors simultaneously, with any allocated small errors. Local finiteness then permits such choices for a locally finite list, including summable errors. The map moves a point only inside its own simplex, so multiplication by simplex suprema of continuous weights proves the weighted upper bound. The same reasoning applies to every regular parameter branch, and hence respects multiplicity. ◻

Proof. Each area-line hit is an isolated intersection of the image of a regular source surface with an affine target interval. Apply Lemma 16, then Corollary 17, in a paired source and target neighborhood. For a curve-plane hit, interchange source and target and clean the preimage plane. The supports can be chosen disjoint and locally finite. Area-line hits avoid curve-plane hits and the measured curve images; curve-plane changes can avoid all area-test lines. If parametrizations share a regular source stratum at a hit, the operation is performed once on that stratum and its fixed multiplicity is retained. Lower strata and other sheets are absent there by genericity. A surviving hit remains in its intended source stratum and target tie simplex, and the cleaning does not increase its count.

Use Lemma 21 to obtain a locally bi-Lipschitz map \(G\) with finite closed-piece \(C^1\) formulas, keeping smaller exact crossing and center neighborhoods. Take the approximation smaller than the positive gaps from all forbidden strata off those neighborhoods. The measured intersections of \(G\) are now regular transverse intersections and their counts do not exceed the original budget counts. Choosing all supports and approximation errors finely also retains the stated compact forward and inverse conditions. These choices use only topology and qualitative approximation; no derivative bound for \(G\) is charged to a budget.

Apply Lemma 25 in the target and set \(H_0=T\circ G\). It has the asserted local regularity and is an onto homeomorphism. On each compact measured list choose the floors high enough that the image measurements cost at most the flat weights plus their allocated errors. Finitely many such lists meet each simplex, so the uniformity in arbitrarily higher floors permits simultaneous choices for the locally finite family.

For the relative assertion, first keep each protected-germ modification deep inside its exempt region. Make the mesh so fine that all simplices touching the protected no-movement core, and their needed neighbors, lie inside that exemption. Assign the value one to their \(b_\sigma\) constants; then \(T=\operatorname{id}\) on a smaller neighborhood of the germ. All simplices required for measured points, including points where the final weight is nonzero, remain buffered away from these cores and use the original \(f\)-crossings. Subsequent cleaning supports are disjoint from the core neighborhoods. Every motion by \(T\) stays inside a simplex. Thus the chosen fine mesh retains the strict margins and the reciprocal compact accuracies, and \(w(TG(x))\leq\sup_\sigma w\) whenever \(G(x)\in\sigma\). This proves the weighted estimate as well. No unweighted estimate through an exempt affine-star core is asserted or needed. ◻

Proof. Use the joint variables \((t,z)\) of (37) for a candidate tetrahedron. For each fixed \(z\), its dual \((3-m)\)-piece \(Z_z\) has diameter \(O(d)\) and measure at most \(Cd^{3-m}\); translation by \(t\) produces the actual dual piece. If \(N(t,z)\) is its intersection count with the measured trace in the candidate’s fixed enlarged neighborhood, translation coarea gives \[\int N(t,z)\,dt \leq C d^{3-m}\, \text{local trace image $m$-measure}.\] Indeed this is the area formula for the difference of the trace and dual-piece parametrizations, whose angle factor is at most one. The joint density bound therefore gives, for every such \(z\), \[\int q(t,z)N(t,z)\,dt \leq C d^{-m}\, \text{local trace image $m$-measure}.\] These are unnormalized translation integrals. Integrating \(z\) over its bounded normalized set only changes the constant. The flat \(m\)-face weight is at most \(Cd^m\) for every \(z\), so it cancels the remaining scale. Discard the Delaunay selection indicator for this upper estimate. Only boundedly many candidate tuples occur locally and their enlargements have bounded overlap, proving (44). The same coarea calculation gives generic avoidance of negligible sets and finiteness of compact counts. A continuous weight’s supremum on each vanishing simplex differs from its value at a measured point by its local modulus of continuity. On a compact set multiply that modulus by the finite trace measure; a locally finite exhaustion and allocated errors give the weighted assertion. ◻

Proof. Discard an arbitrarily small amount of length with multiplicity and decompose the remainder into finitely many compact pieces on \(C^1\) graphs of arbitrarily small slope \(s\) over straight axes. Such pieces may be taken injectively from the regular parameter branches, so multiplicity is accounted for by separate pieces. Cover their axis projections by finitely many intervals of total length at most the sum of their graph-piece lengths plus an arbitrarily small error. Extend the graphs over these intervals with comparably small slopes.

For each fixed relative-coordinate value \(z\) in the joint integration formula (37), classify a dual plane by its unit normal \(\nu(z)\) and the graph axis \(e\). If \(|\nu\cdot e|>2s\), the restriction of its affine defining function to the graph is strictly monotone. Each such plane patch meets the graph over its interval at most once. Charge a contributing tetrahedron to an axis interval of length comparable to \(d\) centered at a hit. Intersecting charged intervals have comparable scales, by slow grading, and their vertices belong to a bounded local candidate list. Thus the charges have bounded overlap. The sum of mesh diameters, and hence of the flat edge weights, is bounded by a constant times the graph length plus the local mesh upper bound at each interval endpoint. After the finite cover is fixed, all endpoint overhangs can be made as small as desired by the mesh choice.

For the remaining normals, the translation-coarea factor on this graph is at most \(Cs\). For each fixed \(z\), integration over the physical translation variable bounds the crossing integral by \(Cs d^2\) times the local graph length. Multiply by the joint density bound \(Cd^{-3}\) and the edge weight \(Cd\), and then integrate over the bounded relative-coordinate set. This gives expected cost at most \(Cs\) times the graph length. The normal classification is translation invariant, so it is legitimate inside this unnormalized joint integral. The omitted trace length has correspondingly small expected cost by Lemma 27. Choose the omitted length and \(s\) sufficiently small, then choose the mesh bounds for the finite graph cover. Markov’s inequality applies only to these small remainders, and makes their failure probability below \(\delta\) without changing the constant on the retained graph length. This proves (45).

For a locally finite family assign summable failure probabilities and errors and impose the corresponding local upper bounds together. A union bound proves the simultaneous assertion. In a deep lattice patch perform the entire argument after the anisotropic rescaling. There are finitely many rational tie-plane families in these units; the same argument, or translated Riemann sums for their projected length coarea, gives the claimed constant. Physical flat weights are bounded by the longest-side factor times their grid-unit weights. ◻

Proof. Rescale the grid to unit cubes, temporarily recording physical flat weights separately. For \(k=2\) slice a cube at a generic height in its thin direction. Each Kuhn tetrahedron projects to a triangle with one repeated vertex. Write the two unsplit projected barycentric masses as \(q_s,q_t\). On a fixed-height slice, either part \(\lambda_0\) of the split mass is a constant minus a sum of some of \(q_s,q_t\), with coefficients zero or one. Consequently \[\det D_{(q_s,q_t)}(q_s-\lambda_0,q_t-\lambda_0)>0.\] These are also the orientation coordinates for a leader triangle with three different projected vertices. Thus every such projected triangle is counted positively.

The total projected weight on a horizontal period is exactly its area. One way to see this without estimating individual positions is to take a constant large \(a\) in (39). Projection of this periodic simplexwise map on the horizontal slice is a torus map with periodic displacement and degree one. Its signed Jacobian integrates to one base area. Lemma 25 converts this integral, in the large-\(a\) limit, into the signed projected triangle weights. Generic heights give only finitely many transverse hits away from simplex boundaries; the signs just computed are positive, and repeated projections contribute zero. For a direct check, the two Kuhn tetrahedra whose vertical step comes first supply total projected area one at heights \(2/3<z<1\), those whose vertical step is middle supply one for \(1/3<z<2/3\), and those whose vertical step is last supply one for \(0<z<1/3\). Each nondegenerate leader triangle contributes \(1/2\) at its applicable height. The finitely many exceptional heights do not affect translation averaging.

In physical units a nondegenerate projected triangle differs from its projected area by at most \(C\eta\) times the base area; a triangle with repeated projections has area at most \(C\eta d^2\). The number of hits per period is uniformly bounded. The total physical area cost is therefore at most \((1+C\eta)d^2\).

For \(k=1\), fix generic coordinates transverse to the long axis and traverse one axial period. Within a Kuhn tetrahedron the varying barycentric pair transfers mass from its lower axial position to its upper axial position, while the other masses are fixed. A switch between leaders with different axial projections is consequently always forward. Two fixed unsplit masses do not tie at a generic transverse coordinate. The projected periodic map has degree one, so total projected length is exactly the axial period. Edges with equal axial projection, and the transverse parts of the other edges, cost only \(O(\eta)\) in period units. This proves the sharp length coefficient.

Translation coarea expresses each coefficient on a trace Jacobian as a finite homogeneous sum of absolute linear forms on its \(m\)-vector. The parallel-vector computations therefore give the asserted tangent coefficients. Splitting a vector into its tangent part and its error and using the triangle inequality gives a finite \(C_\eta\) on the error. A fixed normalization preserving orthogonality of the two subspaces changes these constants only through its two norm bounds. The final length assertion follows by making the anisotropic rescaling before applying Lemma 28. ◻

Proof. Use the construction of Lemma 23, now with cubical rather than anisotropic deep patches, and impose uniform altitude avoidance on the unstructured sites. Conditional on the lattice shifts, enumerate these sites and place them successively. When placing a site, exclude a neighborhood of thickness \(\varepsilon_0\) times its scale around the affine span of every nearby tuple of at most three earlier or lattice vertices. Candidate neighbor lists are fixed and bounded by localization. A neighborhood of each such plane, line, or point removes at most \(C\varepsilon_0\) of the jitter volume. For an absolute sufficiently small \(\varepsilon_0\), at least half the jitter volume remains. Choose uniformly in the remaining portion.

Any affinely independent lattice prefix of a candidate simplex comes from one rotated cubical lattice. There are only finitely many relevant normalized configurations, so its nonzero determinant or lower-dimensional volume has an absolute lower bound. Each subsequently placed vertex has altitude at least \(\varepsilon_0\) times scale over the affine span of the preceding vertices. Induction gives a uniform lower bound on every nondegenerate simplex determinant after scaling. Together with the upper diameter bound this gives uniform shapes. Fully lattice simplices have the usual finite collection of Kuhn shapes. Deep cells may be left cubical, using the same diagonals on shared facets when meeting triangulated cells. Block modifications can consequently be kept inside cubical regions, with whole shared squares on their outer boundaries.

Given the previously placed sites and the lattice shifts, each new jitter has conditional density at most twice its original uniform density. Successive integration of the unselected sites, or the corresponding iterated-expectation bound on rectangular events, therefore bounds the joint density of any fixed tuple of \(k\) selected unstructured sites by \(C^k d^{-3k}\), uniformly in the lattice shifts. The jitters need not be independent. Include the uniform density of a candidate’s lattice shift, if present, and make the same change of variables as in Lemma 23. This again gives a joint density \(q(t,z)\leq Cd^{-3}\) for physical translation and normalized relative coordinates. Discard any later selection indicator. For each fixed \(z\), an \(m\)-face has area \(O(d^m)\), and Fubini gives \[\int dt\int_{t+F_z}g\,d\mathcal H^m \leq Cd^m\int_{U}g,\] where \(U\) is the candidate’s comparable enlarged neighborhood; only translations in its bounded support are included on the left. Multiplying by \(Cd^{-3}\), then integrating \(z\) over its bounded set, gives \(Cd^{m-3}\int_U g\) for the expected face integral. Its mesh scale is comparable to \(d\) by shape control. Multiplication by \(\ell_F^{3-m}\), followed by bounded candidate counts and overlap, proves (46).

For grouping fine cubes into macro-cubes, choose independently a uniform fine-period shift and a uniform discrete grouping offset. Their sum is uniform over the macro-period. Truncation of a lattice list concerns sites inside its fixed buffered patch; it does not translate the transition-site lists by many mesh steps. Thus every mixed local candidate still uses only one fine-scale physical translation variable after subtracting that shift and normalizing its jitter coordinates. Its bounded joint density, and hence the unnormalized joint Fubini calculation, remain unchanged. ◻

Proof. For each \(j\ge0\) and each \(\nu\), apply the density theorem on the parameter piece to \(q_j^{-1}(C_\nu)\). At almost every point of this preimage, locality of the weak derivative gives \[D(v\circ q_j)=Dv_\nu(q_j)Dq_j.\] Remove the union of the exceptional sets for the countably many \(j\) and finitely many \(\nu\). At a remaining parameter point, every branch containing \(q_0\) gives the same product with \(Dq_0\), namely \(D(v\circ q_0)\). If a branch occurs along infinitely many \(q_j\), its closedness implies that it contains \(q_0\). Continuity of its derivative and convergence of \(Dq_j\) therefore show that its products converge to this common value. Finiteness of the branch list proves pointwise convergence of the full sequence. The fixed derivative bounds give a bounded majorant, so dominated convergence proves (48). The proof takes place on \(D\); it does not pull back an ambient null set through a possibly rank-deficient map. ◻

Proof. We give the selection, the maps, and the estimates in their order of use.

Proof. Sobolev maps are approximately differentiable almost everywhere. Truncate the positive-rank sets where their nonzero singular values lie in \([M^{-1},M]\), and choose \(M\) so large that the lost derivative integral is small. Rank-zero points have zero derivative energy and need not be included. On each retained part, pass to compact subsets with continuous jet data and uniform approximate differentiability, losing an arbitrarily small further derivative integral.

These are \(C^1\) Whitney jets after a further compact refinement. To see the value condition, take two nearby retained points and compare their approximate first-order expansions at a common good point in the overlap of balls of radius comparable to their separation. The uniform density losses can be made smaller than a fixed fraction of the overlap, so such a point exists. Subtracting the two expansions, using continuity of the jets, and then sending the differentiation error to zero gives the Whitney remainder estimate. The \(C^1\) extension theorem (Whitney 1934, pt. I, Section 4, Theorem I) supplies the extensions. At full rank, continuity of the derivative and the inverse function theorem give local invertibility. Injectivity on the compact set, together with compactness, makes the extension injective on a sufficiently small neighborhood of the whole compact piece. Its derivative and inverse derivative bounds can be kept within \(C_M\).

For ranks one and two, cover the bounded rank-\(k\) matrix range by finitely many \(\delta\)-balls centered at rank-\(k\) matrices with comparable nonzero singular values. Refine the compact pieces accordingly, losing an arbitrarily small integral to make the finite pieces disjoint. Their number does not enter the local chart bounds. Injectivity of \(f\) makes their compact images disjoint as well. Regularity of the integrable measure \((1+|Df|^p)\,dx\) gives (60); continuity of \(f^{-1}\) permits target buffers whose preimages stay inside these source neighborhoods. The initial losses can be allocated to give (58). ◻

Proof. Translation averaging on the fixed product templates, with block weight \(h^3\), bounds the expected sum of normalized trace errors by \[ C_l\sum_i\int_{N_{Ch}(K_i)}|Df-D_i|^p. \tag{66}\] The measured curve strata, and the measured surface strata when \(p=2\), are covered by Lemma 22. For the additional unprojected rank-two plane section when \(p<2\), ordinary Sobolev slicing and translation Fubini supply the \(W^{1,p}\) trace used in these tests. This step requires neither an image-area formula nor a two-dimensional Lusin property. On \(K_i\), the integrand is at most \(\delta^p\); off \(K_i\) it is bounded by \(C_M(1+|Df|^p)\), whose excess integral is small. Markov’s inequality therefore makes the failed volume small relative to the fixed threshold \(\tau^p\).

There are \(O(l^{-2})\) shell cells of side \(lh\). In grid units, with the longest side restored after the anisotropic rescaling, (63) prevents inflation of the model term. The factors from \(DP_0\) and from anisotropy multiply only \(Df-D_i\). Lemma 28, followed by Hölder on each curve, gives the edge length-power budgets. At \(p=2\), Lemma 27 and the squared derivative trace integrals give the face budgets. A bounded model derivative costs \(C_M(lh)^3\) per shell cell after conversion; multiplying by \(O(l^{-2})\) gives \(C_Mlh^3\) per macro block. For fixed \(l,\eta\), the error contributions are made as small as desired by \(\tau\) and the budget-transfer tolerances. All relevant images lie deep inside their prescribed target patches for small \(h\), by the source and target buffers. This proves (65). ◻

Proof. First reparametrize every unprescribed edge at constant speed, keeping common vertices. On an unprescribed face, cone-extend these edge changes. If \(p<2\), Lemma 15 in dimension two gives face energy bounded by \(CH\) times its boundary \(p\)-energy plus a chosen error. If \(p=2\), use the crude estimate of Lemma 14; after scaling it bounds the face energy by a constant times its base image area plus \(H\) times its edge squared energy. Shape-controlled polygons are converted to square parameters by fixed bi-Lipschitz closed-piece smooth charts. Prescribed faces are left with their given parametrizations. Make one choice per shared face.

The disk hypotheses follow from the actual base and edge regularity. Restricting finitely many closed-piece \(C^1\) formulas gives the almost-everywhere essential limits on a base face and the tangential limits at its parameter edges. On each fixed edge subdivision, normalized arclength has derivative with essential limits almost everywhere. Bi-Lipschitzness bounds that derivative away from zero, and the inverse arclength map has the same property.

For prescribed kernel edges, the source self-map is the restriction of \(T_B\). In the rank-one case it is the interval arclength change; in the critical rank-two case it is the arclength change on each fixed \(J\)-subinterval of the section edge. The disk parametrizations fix the tile boundaries, so they leave these edge changes intact. Kernel-axis edges use the identity. Reusing these source self-maps at positive \(\lambda\) retains their essential derivative limits, even though the resulting curves need not have constant speed for the base map \(w_\lambda\). On radial-core facets the prescribed map already equals \(v\), so \(s_F=\operatorname{id}\). Thus all prescribed edges needed for a critical face satisfy the stated condition. Coning the edge changes preserves the compatibility required by Lemma 14.

The assembled boundary map \(g\) gives the face-preserving bi-Lipschitz self-homeomorphism \(v^{-1}\circ g\) of \(\partial D\). Apply Lemma 15 to this boundary self-map in dimension three, where \(p\le2<3\). It contributes a factor \(H\) times the true boundary-face energy. Substituting the preceding face estimates gives (78). This last collapse requires the radial compatibility of the base parametrization and a fixed bi-Lipschitz boundary map; it does not require stronger smoothness of the entire boundary reparametrization. Allocate the intermediate errors so their converted sum is below the prescribed cell error. ◻

Proof of Theorem 32. We first construct locally bi-Lipschitz approximants; smoothing is performed only after their global derivative costs have been estimated. If necessary, compose with a reflection to fix an orientation convention, and undo it at the end.