Positive scalar curvature and uniform codimension-two width

Proof. Gromov’s comparison, in the form recorded by Nabutovsky, Rotman, and Sabourau [32, Definition 6.1 and equation (6.1)], uses the same continuous-map and entire-fiber width convention as above. A target of dimension at most n-2 is also admissible for \UW_{n-1}. Since M is closed, the comparison and the scaled form of Theorem 1.1 give

□

Proof. It suffices to consider a connected manifold; passing to its orientation double cover if necessary preserves the hypotheses. Suppose such a metric exists. Its scalar curvature has a positive minimum, so the lifted metric on the contractible universal cover X is complete with a uniform positive scalar-curvature lower bound. The deck group acts isometrically and cocompactly. The scaled form of Theorem 1.1 gives a continuous map from X to an (n-2)-complex with uniformly bounded entire fibers. The width and filling-radius implications in Gromov’s noncompact argument A^+\Rightarrow B [20, §3, pp. 6–8] exclude such a map on a contractible universal cover with a cocompact deck action. This gives the contradiction.□

Proof. We construct F by prescribing its inverse in the first variable. For the moment, let k>0 and let g be a positive smooth function on [0,\infty), both to be chosen. Write f>0 for an independent pressure value and, for t>0, set

Put A(y)=g(y)+2yg'(y). Differentiation at fixed f or at fixed t, respectively, gives

Thus A>0 will make X invertible in f. For the inverse, F_x=1/X_f and -FF_t/F_x=fX_t. We will arrange the favorable pressure branch on a decreasing part of g, and obtain the quadratic branch from a lower bound for t^2X_f. We choose g to decrease for small y and to equal y for large y, while keeping A positive through the transition.

We construct g as a smooth rounded maximum with a controlled negative derivative. Fix a nonnegative smooth probability density \rho supported in [1,2], positive in (1,2), and vanishing to infinite order at its endpoints. For example, normalize

Write \mu=\int_1^2r\rho(r)\,\dd r, so 1<\mu<2. For D_g>0 set

Here and below smoothness at y=0 means smooth extendibility to a neighborhood of [0,\infty). To verify smoothness, put

Then M''=\rho, with \rho extended by zero, and g(y)=h(y)M(y/h(y)). Thus g is smooth; moreover M(z)=\mu for z\le1 and M(z)=z for z\ge2. Consequently g>0, g\ge y, and g(y)=y for all sufficiently large y.

For a direct derivative estimate, let

At z=y/h(y) the identities

give

In particular the lower bound for A/g is numerical and does not deteriorate as D_g increases.

We now specify the order of all choices. Choose

Choose D_g only after k,Y, large enough that

Since y(1+y)^{1/4} is increasing, y/h(y)\le1 on [0,Y]. Hence throughout this interval

The equality cases at the endpoints are covered by the smooth matching already proved.

With this g fixed, define the finite constants

For example,

because g(y)\le\max\{y,2D_g(1+y)^{-1/4}\}. Finally choose

Thus the order is k, then Y, then D_g and g, and then T; none of the preceding choices depends on a later one.

We now invert the map X in Equation (2.2). By Equation (2.5), X_f>0. As f\downarrow0, the argument y tends to infinity, where g(y)=y, and X(f,t)\to0. As f\to\infty, y\to0, g(y)\to g(0)>0, and X(f,t)\to S_0:=\pi/(2k)\le1. Thus for every t>0, X(\cdot,t) maps (0,\infty) bijectively onto (0,S_0) with positive derivative. Its inverse f=F(x,t) is smooth by the implicit function theorem.

On a neighborhood of f=0 the exact identity g(y)=y gives

Define F(0,t)=0 and extend F oddly to -S_0<x<0. The displayed formula proves joint smoothness across x=0, including at the endpoint t=T by smooth extension to t>0. The inverse construction also proves the asserted pole limits. For uniformity, if I\subset[T,\infty) is compact and H>0, then \max_{t\in I}X(H,t)<S_0. Whenever x exceeds this maximum, monotonicity gives F(x,t)>H for every t\in I. The negative pole follows by oddness.

It remains to prove Equation (2.1). For x>0 set f=F(x,t) and use the variables in Equation (2.2). Implicit differentiation of X(F(x,t),t)=x yields

In particular the sign in the last identity is the sign of g'. Since A\le g+2y\le3g and y\le g, we have

This proves F_x\ge kt\ge A_0t without a constant depending on g. Also |yg'|\le g: for g'\ge0 this follows from g'\le1 and y\le g, whereas for g'<0 the lower derivative bound in Equation (2.5) even gives |yg'|\le g/4. Therefore

For the second inequality a negative numerator is already favorable; for a nonnegative numerator use g'\le1.

Consider first y>Y. Using A\ge g/2\ge y/2 and g/y\le B gives

The last inequality uses Equations (2.6) and (2.9). This bound remains uniform as y\to\infty, and it proves F_x\le b_0t^2 throughout this region.

If 0<y\le Y, then q=t/y\ge T/Y and

It follows from Equations (2.8) and (2.3) that

The two regions therefore establish the alternative in Equation (2.1). In this second region we also have

so F_x\le\eps F^2. Together with the small-F_x bound for y>Y this proves the final inequality of Equation (2.1).

Oddness preserves all five inequalities on x<0, since F_x, F^2, and FF_t are even and |F_t|,|F| are even. At x=0 they follow by continuity; more explicitly, F=F_t=0 and F_x=kt, while T\ge12kB^2/b_0\ge k/b_0 gives the small-F_x branch there. All claims follow.□

Proof. For S<S_0 the functions u_\pm=\tau\psi\pm S\eta have uniformly bounded first and second derivatives, independently of r,\tau,S. Consequently the left side of Equation (3.14) at these functions is bounded. Since J\ge as_0>0 and t(N) is compact, the uniform pole limits of F permit one choice S_1<S_0 for which u_+ has strictly negative residual (left side minus right side) and u_- has strictly positive residual. Comparison at a maximum or minimum, using strict monotonicity in u, gives the height estimate.

One can make this argument before a solution has been constructed by extending F smoothly to all first arguments with F_x>0, retaining its original values on a slightly larger interval than [-S_1,S_1]. For example, extend the positive even function F_x by a smooth cutoff to a positive constant and integrate from zero. Solve this globally defined equation. The same barriers force every solution into [-S_1,S_1], so that the extension is never used in the resulting solution.

We give the gradient estimate, including the dependence of the pressure on the graph angle. If \max_N|\dd u|_h\le1 there is nothing to prove. Otherwise, near a point of maximum slope, \mu=1 and the graph in (N,h)\times\R is an ordinary graph. Write

for its downward unit normal. Denote its second fundamental form and mean curvature by \mathrm{II} and H, with H=\Div\nu. Extend t constantly in the vertical direction and put X=-\grad t and K=-\partial_z. The equation is

Let K^\top=K-w\nu. Under a normal variation with speed v,

Thus the linearization of the equation, with all terms moved to the left, is -\mathcal Lv, where

The last derivative differentiates the base and height variables while holding the angle fixed. Translation in direction K leaves the angle, H, and \langle X,\nu\rangle unchanged and differentiates the residual by p_z. Its normal speed is w; its tangential component differentiates an identically zero residual. We obtain the exact identity

In particular the term -p_\omega K^\top in the drift has not been discarded.

Put \xi=(u-\tau\psi)/\eta. On the graph,

The height bound puts F,F_x,F_t in a fixed compact range. Moreover |N_u|\le1, |\dd\psi|_h\le1, and |\dd\eta|_h\le s_0/4. The zero-order coefficient in Equation (3.16), after omitting the nonnegative term |\mathrm{II}|^2, is therefore at least -C/\eta, with C independent of r,\tau. At the minimum of w under consideration, the slope is at least one, so J\ge1/\sqrt2 and p_z\ge A_0T/(\sqrt2\eta). Since \grad w=0 and \lap w\ge0, Equation (3.17) implies

This gives a positive lower bound for w, and hence the asserted G.

We finish the fixed-r continuation argument in enough detail to specify its regularity inputs. A classical C^{2,\alpha} solution is smooth before we apply estimates requiring smoothness. Indeed, writing its equation as a^{ij}u_{ij}=f, the coefficient functions a^{ij} and f are C^{1,\alpha}. On nested coordinate balls, a difference quotient u_h satisfies

For each nonzero h, u_h is already C^{2,\alpha}; its right side has a C^{0,\alpha} bound independent of h. The classical interior Schauder estimate and passage to h=0 give u\in C^{3,\alpha}, and iteration gives smoothness. These qualitative estimates may depend on the existing C^{2,\alpha} norm; the following a priori estimates do not. At \tau=0, u=0 solves the equation. For fixed r>0, the height and gradient bounds put (x,u,\dd u) in a compact set on which the principal coefficients are uniformly elliptic and all explicit coefficient derivatives are bounded. In a coordinate chart write the divergence-form equation as

Differentiating with v=\partial_ku gives

The x derivatives on the right are explicit derivatives of the coefficient functions. All nondifferentiated terms and the vector field inside the last divergence are bounded using only the compact jet range; no second derivative estimate has been assumed. The local Hölder estimate for uniformly elliptic divergence-form equations, with bounded lower-order coefficients and bounded divergence data, now gives a uniform C^{1,\alpha} bound for u on a finite cover. The equation in nondivergence form then has Hölder coefficients and right side, so local Schauder estimates give C^{2,\alpha} bounds, followed by bounds of every order. The Hölder and interior Schauder estimates used here are respectively Simon [37, Lecture 18, Theorem 3, and Lecture 12, Theorem 2]. All their uses are on compact coordinate balls with a positive ellipticity constant; their constants may depend on fixed r.

Fix the floor r>0, and choose 0<\alpha<1 smaller, if necessary, than the exponent in the preceding Hölder estimate. The linearization in u has strictly negative zero-order coefficient -JF_x/\eta; angle and gradient derivatives affect only its principal and first-order terms. We verify that this linearization is an isomorphism from C^{2,\alpha}(N) to C^{0,\alpha}(N).

First consider \lap_h-1. For f\in L^2(N), minimize on H^1(N) the strictly convex coercive functional

Its Euler equation is (\lap_h-1)v=f in the weak sense. Testing the homogeneous difference with itself proves uniqueness among H^1 weak solutions. For f\in C^{0,\alpha}(N) choose smooth f_j converging uniformly to f, with uniformly bounded C^{0,\alpha} norms, using coordinate mollification and a finite partition of unity. Weak interior regularity makes the corresponding weak solutions v_j smooth [37, Lecture 6, Theorem 1 and Corollary 1]. The maximum principle gives \|v_j\|_{C^0}\le\|f_j\|_{C^0}. A finite interior cover and the Schauder estimate [37, Lecture 12, Theorem 2] then bound v_j uniformly in C^{2,\alpha}(N). Passing to a subsequence in C^{2,\alpha'}(N), where 0<\alpha'<\alpha, gives a solution for f. The common \alpha-Hölder bound for the second derivatives passes to the limit, so this solution belongs to C^{2,\alpha}(N). Weak uniqueness identifies it with the energy solution, and the maximum principle also gives uniqueness in the classical class.

Let L_q, 0\le q\le1, interpolate the coefficients of \lap_h-1 and the linearization written in nondivergence form. Uniform ellipticity and a uniform strictly negative zero-order bound persist along this path. The maximum principle and the same finite-cover Schauder estimate give

Invertibility is open by the Neumann series. To see closedness, suppose q_j\to q through invertible parameters and, for a fixed f\in C^{0,\alpha}(N), put v_j=L_{q_j}^{-1}f. The estimate above bounds v_j and gives

Thus v_j converges in C^{2,\alpha} to a solution of L_qv=f; injectivity follows from the same estimate. This proves the required linearized isomorphism.

The implicit function theorem now gives openness of the nonlinear continuation in \tau. The preceding difference-quotient bootstrap applies to the C^{2,\alpha} solutions obtained by implicit inversion. The uniform a priori C^{2,\alpha} bound makes its higher derivative bounds uniform along continuation at this fixed positive floor. In particular these bounds give compactness in C^{2,\alpha}(N), so a convergent sequence of soluble parameters has a solution at its limit. The height bound keeps the pressure argument in its fixed compact interval. The soluble set is therefore closed as well as open and contains 0, hence equals [0,1]. The local bootstrap makes the resulting solution smooth.□

Proof. Here \eta=r and |u_r/r|\le S_1. Expanding the left side of Equation (3.14) and using F(0,t)=0 writes its equation as

The coefficients a^{ij} are uniformly elliptic and bounded, and b^i are bounded, independently of r. Indeed they are smooth functions of (x,Du_r) on the fixed compact jet range from Equation (3.15); the explicit x part of the flux vanishes at zero covector and is therefore a bounded first-order term. Also c_r is bounded above independently of r. No derivatives of these coefficients are needed for the next comparison.

Take a coordinate box compactly contained in the constant-floor region. Let \ell_f be its inward affine distance from face f in these coordinates and set

where M_0 dominates |u_r| on the boundary for every r. Let \Lambda bound a^{ij}(\partial_i\ell_f)(\partial_j\ell_f), let c_*>0 be the lower bound for c_r, and let B_* bound the first-order contributions. The operator in Equation (3.19) applied to each summand of V_r is at most that summand times

Choose \gamma>0 with \Lambda\gamma^2<c_*/2. For small r this quantity is negative. Comparison with both u_r and -u_r proves, on each smaller box, |u_r|\le C\exp(-c/\sqrt r). A finite covering gives this estimate on any fixed compact subset.

To upgrade the estimate to derivatives, rescale a coordinate ball of radius comparable to r by

The equation becomes

where \mathfrak B_0 includes the drift term and JF(v,t) and has no negative power of r. We have |v|\le S_1 and |Dv|\le G. The principal ellipticity is uniform. Explicit y derivatives of \mathfrak A have a factor r, explicit y derivatives of r\mathfrak B_0 have a factor r^2, and its v,p derivatives have a factor r. Thus all explicit coefficient derivatives are bounded uniformly on this fixed jet range. Apply the Hölder estimate to the differentiated Equation (3.18), followed by the Schauder bootstrap on fixed balls in y. For every k this gives

on smaller balls, with constants uniform in x_0 in a fixed compact set and in small r. Finally, interior interpolation between the exponentially small C^0 bound and the polynomial bound for one higher derivative proves C^k convergence to zero for each fixed k. For example, on fixed nested compact subsets of the constant-floor region, interpolation bounds the third derivatives by a constant times r^{-9/4}\exp(-c/(4\sqrt r))+\exp(-c/\sqrt r), which tends to zero. This last case ensures C^2 convergence of all metric and drift increments in Equation (3.7), which are smooth functions of Du_r.□

Proof. We derive the scalar comparison explicitly. Near a point with \dd u\ne0 use q=u as the transverse coordinate and write

Here \mathrm J is the second fundamental form of the level sets, and all tangential norms and derivatives below use \gamma_q. Put S=H-et. The Gauss equation and normal variation of H give

Consequently

The deformation replaces v by v/l and t by t-\beta. The change in the tangential terms on the right is

This computation follows by expanding \lap_\gamma(v/l)/(v/l) and |\dd_\gamma(t-\beta)|^2; the \lap_\gamma\beta and the terms involving \dd_\gamma t cancel.

The divergence equation can be written

Define

Because e\log\sqrt\theta=-K, we have S=M+K. For the new metric, e'=le, \mathrm J'=l\mathrm J, and

Substitution in the normal terms of Equation (3.22) gives

For completeness, the derivative terms before simplification are -2le(lM+K/l)+2e(M+K). Their eK terms cancel; using e\beta=\theta K/l^2 leaves 2\theta eM-2\theta MK+ 2\theta K^2/l^2, which combines with the squared terms to yield Equation (3.23).

Since e\chi=-m\chi K and et=H-M-K,

Completing the H and K squares gives E\ge\mathcal B, where

An exact verification, useful for tracking every sign, is

All pressure expressions here are evaluated at (\xi,t).

Suppose first that s\ge s_0. Denote the expression in braces in Equation (3.24) by \mathcal C. Its F^2 coefficient is nonnegative, and 1\le1+l^2(1+m)\le2. From Equation (2.1) we get

Indeed, if FF_t\ge0, then F_t^2\le\eps FF_t and \mathcal C\ge-(4+n\eps)FF_t\ge-5\eps F_x. If FF_t<0, the same absolute-value estimate gives \mathcal C\ge(2-n\eps)|FF_t|\ge|FF_t|.

Where \eta\ne1 we have \psi=0, and

Using 1\le\chi\le2, Equations (3.9) and (3.25) give E\ge sF_x there. Where \eta=1, we have e\xi=s-e\psi\ge s-b, with b^2=h_0(e,e) as in Equation (3.8), because |\dd\psi|_{h_0}\le1. If s>8b, the same estimates give

It remains to consider s_0\le s\le8b. The possible deficit from sF_x is at most 10F_x: indeed 2\chi(s-b)-s\ge-4 and 20\eps<1. On the favorable pressure branch, however,

since s\le8b\le8. Thus a deficit is possible only when F_x\le b_0t^2. In that case b\ge s_0/8 and our choice of b_0 gives 10b_0\le\delta(s_0/8)^2. We conclude throughout s\ge s_0 that

The last inequality follows from F_x\ge1000t and s\ge.01. Combining Equations (3.6), (3.23) and (3.8) proves the raw inequality in this region, even without its d term.

Now suppose 0<s<s_0. Since m\le-1/2, the first coefficient in \mathcal C is at least 3/4. Using |F_t|\le\eps|F| gives

Where \eta=1, e\xi/\chi\ge-1. Elsewhere, a negative e\xi can occur only if s\le|\dd\eta|_h, and then

By the quadratic pressure upper bound,

Since \theta\chi^2\le2a^2s_0^2, the loss in Equation (3.23) is at most 2C_2a^2t^2. Also -\beta\le\theta\le a^2s_0^2 and t\ge1, so t_u^2-t^2\le3a^2t^2. The choice in Equation (3.10) pays both terms with (d/2)t^2. The trace penalty is favorable and may be omitted in this regime. This proves Equation (3.21) wherever s>0. At a zero of \dd u it follows by continuity unless \dd u vanishes on a neighborhood; on such a neighborhood the raw output equals the input and the inequality is immediate.□

Proof. Enumerate the labels and choose d_i>0 with \sum_i d_i<\delta. Apply Theorem 3.1 in this order, keeping h_0 as the background metric throughout. Earlier changes increase distances, so every pair still meets the separation hypothesis when it is processed. If (h^{(j)},t^{(j)}) denotes the data after j changes, write

Thus Q_0=n, 0<Q_j\le n, and A_j\le1. We claim that

It holds at j=0. In a single step, abbreviate the old coefficient by A, the new one by A', and the old and new functions by t,t'. The local estimate gives

The last inequality uses both A\le1 and t'\ge t\ge T>0. This proves (4.5). Its coefficient is greater than 1-(n+1)\delta, which proves (4.4). Each processed separation satisfies (4.3), and later changes preserve it.

The supports lie in the locally finite family (K_i). Hence every point has a neighborhood on which the sequence is eventually constant; its limit is smooth and satisfies the same inequalities. Completeness follows from h_1\ge h_0: an h_1-Cauchy sequence converges in h_0, and near its limit the two smooth metrics are equivalent.□

Proof. Let E_\alpha=e^{-p_\alpha}\partial_{\theta_\alpha} be the unit vertical fields. For horizontal fields X,Y, the Koszul formula gives

and \widehat\nabla_{E_\alpha}E_\beta=0 for \alpha\ne\beta. Therefore, for a horizontal unit vector X, the mixed sectional curvatures are

Summing twice over all orthonormal two-planes gives the identity.□

Proof. A complete connected Riemannian manifold is proper. Choose a maximal D-separated set of centers. Maximality gives the covering by closed D-balls. A compact set contains only finitely many centers; more generally, centers whose closed 2D-balls meet a fixed compact set lie in a fixed compact neighborhood of that set. This proves local finiteness of the outer balls and countability of the net. The triangle inequality gives \dist_g(A_i,B_i)\ge D. The last assertion follows immediately if M=B_g(x_i,2D).

We keep a scalar reserve of 1/2 in the original units. Put R_j=L_*^jR_0, start with h^{[0]}=g, t^{[0]}=T, and work in round j with the metric h^{[j]}/R_j^2. At the start of that round we require

For j=0, this follows from \Scal_g\ge1 and R_0^2/2\ge T^2.

A label remains active until its separation exceeds its prescribed L_i. At the start of round j, every active pair has separation at least WR_j in h^{[j]}; this is initially true and follows inductively from (4.3). Apply Lemma 4.1 to the active pairs in the working metric, with reserve c=R_j^2/2. The required support condition follows from \dist_{h^{[j]}/R_j^2}(A_i,B_i)\ge W: the open distance-W neighborhood of A_i misses B_i, and is therefore contained in the original outer ball B_g(x_i,2D). We may take K_i=\overline B_g(x_i,2D).

Return the output metric to original units and denote it by h^{[j+1]}. It dominates h^{[j]}. Scalar curvature, the Laplacian and squared gradient norms all multiply by R^2 when a metric is divided by R^2. Consequently, on passing from working scale R_j to R_{j+1}, the round estimate becomes

By (4.1), the coefficient of the final square is greater than one. Thus (4.8) is restored. In original units the reserve remains exactly 1/2; it has not been spent by the change of scale.

Each fixed label becomes inactive after finitely many rounds, since its separation is at least WR_j as long as it remains active and R_j\to\infty. A compact set meets only finitely many of the original outer balls. Each of their labels is processed only finitely many times. Thus there are only finitely many changes on a neighborhood of any compact set over the entire construction, not merely in each round. The limiting data (h,t) are smooth, satisfy

and satisfy (4.6). The same comparison argument as in Lemma 4.1 proves completeness of h.

It remains to realize the drift term by ordinary finite-dimensional warped products on each compact window. Enumerate the nonzero compactly supported increments of t as \tau_\nu\ge0, so that t-T=\sum_\nu\tau_\nu. The supports are locally finite. For each \nu, choose an integer m_\nu\ge1 such that

The supremum is finite because the support is compact. Use m_\nu fresh log warps, all equal to -\tau_\nu/m_\nu. Listing these functions as (p_\alpha), we obtain

On a neighborhood meeting only finitely many supports, equip a finite product of circles with metric h+\sum_\alpha e^{2p_\alpha}\dd\theta_\alpha^2, omitting all unused unit factors. Its scalar curvature is the left side of (4.7); this is the identity in Lemma 4.2. Substitution gives \mathcal D(h,t)-\sum_\alpha|\dd p_\alpha|_h^2\ge1/2-1/10=2/5. This proves (4.7).□

Proof. In an exact product tail all label memberships pull back unchanged, whereas the face dimension is m-d. A count at least m-1 is at least (m-d)-1, proving the first assertion. The same argument with thresholds m-2 and (m-d)-2 proves the guard assertion. On a new side, whose dimension is m, a point of D_A has at least m-1 remaining undecided labels. This puts its projection in the guard and, because G'_j\subset G_j, in the old required set. On a hit packet, whose dimension is m-1, the required count is m-2. Its projection lies on L\subset G_{j_0}, so the processed label contributes one further old undecided label. The old count is therefore at least m-1. The cylinder projection has exactly the same remaining memberships as its base, so the side argument also covers all cylinder points. These are pointwise count statements and use no scalar estimate away from D_E.□

Proof. Realize the processed profile by a smooth finite pressure and take a local absolute minimizer of weighted perimeter minus its pressure-volume term, as in Lemmas B.1 and B.3. The phase has the forced signs of Lemma B.4; its boundary, called the true cut, has the regular-part geometry of Theorem B.6 but may still have singularities. After the radial cut, the generic pressure choice in Proposition B.5 makes these true cuts converge to the corresponding face products at packet ends (Corollary B.7). On the initial radial parent use shift zero and a phase from compactness; compact barriers replace this post-radial growth and uniqueness argument.

The wall, and each retained side with its attached cylinder, carry the quadratic-form lower bounds of Proposition B.10. These bounds include a positive gradient reserve and, on a joined side, the scalar contribution of its internal attachment seam. They are not yet pointwise scalar bounds on smooth new packets. Keep these true cuts and their pressures fixed. The constructions of Lemma C.1 and Propositions C.5 and D.6 change only the remaining profiles, enlarge their cores, and add nonnegative potentials. Their form cost is paid by the gradient reserve and the gap \lambda_r-\lambda_{r+1}, and the updated guard H_E^* of (5.7) misses the singular set. Thus the future required regions can be controlled through regular portions of the true cut. When the parent dimension is at most seven the singular set is empty, and no singular forcing is needed; compactness on the resulting smooth windows is the smooth case of Lemma D.2. More generally this smooth-window case applies whenever the limiting wall is smooth, including products of smooth faces with Euclidean factors even when their total dimension is at least eight. At the first radial split the same arguments are applied on the locally finite compact transition blocks, as specified in Proposition D.6.

The guarded approximation in Lemma E.2 now replaces the true cut by a smooth regular wall. Its defining functions are exact products in the old ends. The comparison with the true cut on the guard, followed by the seam smoothing of Lemma E.4, transfers the form bound to the smooth split. The positive supersolutions of Lemmas E.7 and E.9 then supply 399 compatible new circle factors; the scalar identity in Proposition E.10 converts that form bound into the pointwise stage-(r+1) scalar inequality. Throughout this transfer the parent scalar inequality is used only over D_E: Lemma 5.3 places every new required region inside that old region as well as inside the guard.

Lemmas A.1 and A.2 give the new geometry; Lemma 5.4 gives the radial confinement. Unit entries are fixed by restriction, product extension, and seam smoothing. For a remaining label passive in the sense stated above, we omit its forcing updates, and Lemmas E.5 and E.6 reproduce its chosen constant potential exactly. The choice of \eta_r satisfies the strict stage margins in (E.3).□

Proof. For a positive number p, the decreasing crossing

from +\infty to -\infty has length

Extend it by its poles. For center i, begin this crossing at stretched distance 2 from A_i and compose it with \dist_h(A_i,\cdot). The distance is 1-Lipschitz, so the resulting profile obeys the inequality with .5 in place of a_0=.60. Take cores

Their closures lie in the respective pole interiors. The stipulated width puts B_i strictly inside the negative core. In particular, on the complement of the original closed ball \overline B_g(x_i,2D), the profile is the negative pole and the negative core is full.

There exists a smooth proper function \rho:M\to\R, bounded below, with |\rho-\dist_h(o,\cdot)|<1 and |\grad\rho|_h\le2. Here no uniform geometric assumption is necessary. To construct it, use relatively compact coordinate balls in a locally finite cover, convolve the distance on slightly larger balls, and blend with a smooth partition of unity. The metric distortion and local convolution errors can be made arbitrarily small on each ball. In differentiating the blend, write each local approximation as the distance plus its error; the terms involving derivatives of the partition sum to zero on the common distance. Choose the value errors so that the remaining partition-derivative terms have sum less than a fixed small number at each point. A locally finite refinement and successively smaller positive errors achieve this, while keeping the total value error below one. Properness follows from the bounded value error and properness of the complete metric h.

Choose alternating positive and negative plateaus in the \rho-axis, with transition intervals wider than 2T(p_0^*). Rescale the preceding crossing to these intervals, reversing its sign on alternate intervals. The gradient bound for \rho gives the profile inequality with coefficient .5 and potential p_0^*. Choose the core closures strictly inside the plateaus. Regular values in the remaining margins give compact smooth barrier annuli, and a compact central barrier, whose closures stay in the same pole plateaus. Their value bands tend to infinity and are locally finite.

A wall respecting the signs cannot cross a separating plateau band; each component is closed in one compact radius block and hence compact. A component of either closed side is trapped between two separator bands of the opposite sign, or in the initial compact block. Regularity of the defining function gives manifold-with-boundary charts for these sides. A compact block therefore contains only finitely many components. The same charts prove local finiteness of the component families. When M is compact, use a radial label with constant positive pole and full positive core; its split has no wall and its only side is compact.

The scalar requirement initially holds everywhere because

Finally a compact block with a compact buffer meets only finitely many original closed outer balls and finitely many initial warp supports. Every other center label and warp has the asserted constant data there.□

Proof. On E, the sum of potentials is exactly

The second term is a finite spatial constant. Indeed

The first term is a finite sum; its derivative bounds follow from compatible smoothness and the compact-part/product-end decomposition in Lemma A.2. On a finite comparison diagram choose common product thresholds for its finite exceptional list. Every other entry is already a spatial constant with the same value on each copy. The full sum is therefore an exact face pullback beyond those thresholds; no supremum of infinitely many thresholds is taken. Passive entries have empty undecided set. Thus a set requiring at least a\ge1 undecided labels is the finite union, over a-element subsets of I_E, of the intersections of their closed G_j’s. This proves closedness, including for the guards with threshold m-2. Threshold zero gives E.

For (ii), take the union of the finitely many relevant old warp indices on the finite diagram and set every missing entry to one. Writing u_\alpha=\log f_\alpha and U=\sum_\alpha u_\alpha, the scalar expression is

A padded logarithm is zero; the product weight is unchanged. The weighted Laplacian is \Delta_{h_E}+\langle\dd U,\dd\,\cdot\,\rangle_{h_E} and is unchanged as well. Weighted perimeter and its first and second variations depend on the same weight. Equivalently, a flat unit factor adds zero second fundamental form and zero curvature to the invariant variation calculation. Thus the wall and seam forms also agree. The finite-diagram geometric lemma now gives completeness, bounded geometry, positive lower and finite upper bounds for the weight, and center-uniform polynomial volume growth. It gives the same statements after taking a finite Euclidean product. The regularity problem for a minimizing phase remains on the base, with singular dimension at most m-8 by Theorem B.6; padding does not alter that dimension.

For (iii), a particular face receives requests from only its finitely many cofaces and coface chains. Allowing for the finitely many fixed product and cutoff bounds, the minimum of the requested positive errors is positive. Settle all these numbers before constructing the approximations. A request only decreases an error tolerance: it does not copy any coface value into the face. During the subsequent upward pass, the inputs on a packet are its own data and already constructed copies from its finite downward diagram. Their exceptional sets have a finite union. Outside that union every copy in (6.8) agrees with every other copy. A newly constructed coface is not used to reconstruct its sources. The same argument applies to finitely many successive directed passes, and proves the stated dependency rule. With these numerical tolerances fixed, at each upward pass the output on E depends only on its original data and the outputs on its strict faces. Its source set for copied functions is therefore contained in the finite downward diagram of E. The preceding downward pass changes only positive numbers, not the input functions. In particular two packets sharing a face do not copy one another’s newly constructed data through that face.□

Proof. We verify the choices in their order of dependence.

The current pressure and minimizing phases. The pressure construction of Lemma B.1 uses a positive floor only for the label being processed. Replace its finite-family floor by p_r^*. Compatible clipping heights can first be chosen above the local barrier bounds, with a uniform extra barrier margin of one. Then use local smoothing errors to obtain

All smoothing requests involve only this one profile and finite incidence diagrams. The elementary inequality

shows that one common nonempty shift interval, for example

preserves both 2|\grad(p_0+c)|\le .67(p_0+c)^2+1.02\Pi_r and the barrier inequalities. Floors for other labels do not enter this interval.

After the radial stage, Lemma 6.3 supplies all bounded geometry and weighted polynomial-growth hypotheses of Proposition B.5. There are countably many packets, Euclidean product dimensions, and tests in that lemma’s exhaustion. Remove the countable exceptional shift sets and choose a single c in (6.11). The resulting true minimizers have the common product limits of Corollary B.7; choose them before any shielding update. At the initial radial step no uniqueness or polynomial growth at spatial infinity is used. The smooth global positive weight and locally bounded smooth pressure give local BV existence and strict compactness by Lemma B.3. The relatively compact radial barriers force the phases by compact comparisons. All variations and tests lie in buffered compact windows with finite warp lists, so the formulas in Proposition B.10 apply. The GMT and shielding dimensions are the base dimensions at this first step as well.

Anchors, guards, and shielding. On each post-radial packet retain its finite exceptional list, enlarged by its finite downward source diagram and by the current label. Labels outside it are already forced everywhere and require no shell addition. Choose compatible smooth clipped anchors as in Lemma C.1, retaining the exact value -1 for a passive profile. For each packet of dimension m\ge8 and each set of m-7 remaining labels, the image of its singular set under the corresponding smooth anchor map has (m-7)-dimensional Lebesgue measure zero. There are countably many such conditions, even if all finite subsets of the countable label set are included. In the countable product of uniform probability intervals (-.3,.3), each bad cylinder has measure zero by its finite-dimensional marginal. A threshold vector outside their union therefore exists. At most m-8 anchors equal their thresholds at any singular point. The sign exclusions at poles follow from the .1 anchor approximation, exactly as in Lemma C.1.

Process the finitely many face depths from deepest upward, keeping the true cuts fixed. Suppose all strict faces of E have been shielded. Its remaining risky set

is closed. It is compact: an escaping sequence would, by Lemma A.3, have a product limit over a further face with compact convergent base coordinates. The finite exceptional list and closed core complements pass the undecided count to that limit. The face has a smaller dimension, so the limit lies in its guard. The product convergence of true cuts forces the limiting point to be singular, since regular-point graphical convergence excludes nearby singular points. This contradicts the completed shielding of that face. This argument also treats escape in several normal coordinates.

Apply Lemmas C.2 and C.3 and Proposition C.5 on this compact risky set, only for the finite list which can still be undecided. Enumerate the signed compact flat pieces. Choose each shell scale by Lemma D.3, hold it fixed, and then send its coface fronts far enough out by Lemma D.4. In the cap construction of Lemma D.5, the crossing for label j satisfies

Its length is finite; the inequality is paid by \Pi_j\ge p_j^*. The minimum-of-branches formula in that lemma adds no simultaneous normal derivative cost. Its cutoff finishes before the cap crossing starts, so whenever the shell branch is active the whole shell potential is available. Opposite signs remain in disjoint strict anchor regions, and cores only enlarge.

Only finitely many additions are required on E. Indeed, every point of the initial compact K_E eventually belongs to an open forced core for each of its strict anchors among the finite exceptional list. At most m-8<m-2 labels can then remain undecided there. The residual risky subsets of the initial K_E are nested and closed, and their intersection is empty. Compactness gives a finite index at which the residual is empty. Stop then. Each specified coface receives additions from finitely many source faces through finitely many chains. Thus only finitely many additions and finite minima or maxima affect any whole packet. In particular no infinite limiting profile is taken on that packet.

To control all the countably many packets at once, enumerate each possible local addition and assign positive allowances with total less than .12. Split each allowance between its shell-scale choice and its simultaneous finite coface extension choices. Starting the bookkeeping form bound at .78, the maximum estimate and the extension estimate give one \theta_*<.9 such that

Here \mathcal N has coefficient 1.7 on a wall and coefficient .8 on bulk plus 1.7 on its attached cylinder. These are the same coefficients on every packet and product. Only a finite list of additions occurs on each post-radial packet, so the bound persists after a whole depth has been treated.

Initially, instead work in separated compact transition shells with compact buffers between radial barriers. Each has finitely many exceptional labels and warps. Its risky set is compact, and the same finite termination applies; there is no coface extension yet. Choose supports within the shell buffers. Their local finiteness makes all updates locally finite on M, and a compact test meets only finitely many of them. The summable allowances prove (6.12) for every such test. Every compact radial side block meets only finitely many shell buffers. This supplies the whole-packet finite lists after the first attachment.

Counts and the quantitative transfer. Core enlargement gives G'_j\subset G_j. Only the finite exceptional list can contribute to the counts on a post-radial packet; the analogous statement holds on buffered compact blocks at the radial step. Thus Lemma 5.3 places every new required set in both the old required set and the shielded guard under projection, including along attached cylinders. The finite lists and closed core complements also pass these count inclusions to all further-face product limits.

The guarded tube and comparison constructions of Lemmas E.1 and E.2 use these closed guards, regular graphical convergence on product limits, and finite packetwise derivative bounds. All are now available. Numerical errors are allocated down coface chains before the defining functions are constructed upward.

Apply the quantitative transfer of Lemma E.3 with the current t_r, \Delta_r=t_{r+1}-t_r>0, and e_0=e_r=\Delta_r/800. Its stage margins (E.3) were proved for these general adjacent parameters, so they do not require equally spaced stages. The potential calculation also applies: on each finite comparison diagram, (6.10) expresses the old potential sum as finitely many smooth functions plus one spatial constant. Only finitely many shielding additions occur there. Thus, with \widetilde\Pi_j=\Pi_j+D_j, the same raw form and comparison estimates control

near the new required set, with the positive mass and gradient reserves proved there. The constant tail has no derivative error; potential approximation errors concern only the finite exceptional list. At the radial step these estimates are made on buffered compact blocks, as in that proof. The seam and potential smoothing can consequently retain an e_r local mass gap. This positive number is fixed for the current stage; all geometric tolerances may depend on the packet. No infimum over future stages or over all packets is taken.

Exact passive constants in regularization. Restrictions, bulk inclusions, and cylinder projections preserve identically constant functions. Seam smoothing of an old identically unit warp leaves it one. For the remaining potentials, smooth (1+3\eta_r/2)\widetilde\Pi_j as in Lemmas E.6 and E.5, so that

On a finite exceptional list, allocate the required relative errors separately, using p_j^*>0 for its jth entry. If an aggregate error is prescribed, subdivide it among that finite list. If an entry is passive on the whole source diagram, set its output exactly to c_{r+1}p_j^*. This is compatible with the smoothing procedure: local convolution of a constant is that constant, partition weights sum to one, and every copied end value is the identical constant. Equivalently this entry can simply be omitted from all smoothing and restored as that constant afterward. We only impose this rule on whole constant inputs and their agreeing copies, so it makes no claim about preserving the edge of an arbitrary nonsmooth plateau.

Before blending on a coface, enlarge its exceptional list by the union over its entire finite downward source diagram. Thus a face-active entry is recorded even if the coface’s individual input was passive. The remaining copies agree identically. Coface requests merely adjust numeric tolerances; they do not introduce new source values on faces. This proves exact passive reproduction on the completed diagram and precludes a cascade through infinitely many same-depth neighbors. At the radial step choose a compact buffer for each compact radial block, enlarged to contain the finitely many chart closures and shell supports used by its approximations. Only finitely many original outer balls and initial nonunit warp supports meet this buffer. Enlarge the exceptional list by precisely these indices. Every other entry is the same constant on every averaging neighborhood and is therefore reproduced exactly before cylinder attachment.

The local-slope argument of Lemma E.6 justifies restriction of profiles to the wall: an almost-everywhere gradient bound by a continuous function gives the corresponding local Lipschitz slope bound at every point of each finite band. Restriction and cylinder projection then preserve it. The small metric change increases slopes by at most 1+\eta_r, and the first inequality in (E.3) together with (6.14) gives the next profile inequality. Continuous pullback and core enlargement preserve all pole-neighborhood and forced-preimage conditions.

Corrections, supersolutions, and the new warp group. The sum in (6.10) is now smooth, bounded, and exactly compatible, as are its finite-order derivatives. On every required finite diagram, pad the old warps by units. In a product end over B the operator of Lemma E.7 therefore satisfies exactly

In particular the potential formula does not depend on the base dimension or on how many unit factors were padded. The correction is zero near the new required set and produces a positive form gap on each whole packet. The proof of Lemma E.7 uses finite minima of the gaps on the packet’s face diagram. Its bounded resolvent and the compatible positive supersolutions in Lemmas E.8 and E.9 apply with the packetwise bounded geometry and polynomial weighted volume already proved. Their tolerances are chosen after the operators and their individual exact solutions are fixed, and before the final upward blending pass. Thus their finite-dependency hypothesis is precisely (iii) of Lemma 6.3.

Let v_A>0 be the resulting supersolution with L_Av_A\ge1/2. Append the common group of k=399 circle lengths f=v_A^{2/(k+1)}=v_A^{1/200}. The scalar identity in Proposition E.10 gives

Only the number of new factors enters the coefficient C. The old finite list may vary from packet to packet; unit padding has no effect. Each stage adds just this one finite group, with the same indices in every status. All its lengths are positive and bounded above and away from zero on a whole packet, and exactly compatible. The initial lists and their finite propagated exceptions remain finite, so the complete old list at stage r+1 is still finite on each whole packet and each finite comparison diagram.

Finally the regular split preserves compact truncations and finite incidence by Lemma A.1. Initially the radial separators make all components compact before prolongation. Compatible smooth metrics and finite relevant warps then give the complete packet geometry by Lemma A.2. The recursion (6.7) preserves confinement in original radial blocks. All exceptional-list enlargements just described are finite, and all passive constants have multiplier c_{r+1}. This completes the induction step, including the first radial step.□

Proof. Such a packet is complete by Lemma A.2 and has only finitely many relevant positive smooth warps f_1,\ldots,f_q on its whole base. Its arclength universal cover is \R, including when the packet itself is a circle. On this line put u_\alpha=\log f_\alpha and P=\sum_{\alpha=1}^q u_\alpha. The warped-product scalar formula reduces to

The required count in dimension one is zero, so the invariant holds everywhere and gives R\ge\lambda_r>0. Consequently

Integration over a line interval longer than 2\pi/\sqrt{\lambda_r} contradicts the range of the arctangent. If q=0, the scalar formula is simply R=0. The argument takes place at one finite stage and uses an ordinary finite sum.□

Proof. Initialization is Lemma 6.1. If the construction has reached stage r, its parents have dimension at least two: a first nonempty output of smaller dimension would have dimension exactly one, since a hit decreases dimension by one and a sign preserves it. Proposition 6.4 produces the next invariant, and Lemma 6.5 excludes that first output. Thus the induction continues through every finite stage. There is no passage to an infinite-stage analytic metric, weight, or torus.

Apply Proposition A.5 to each finite initial segment. Its hypotheses are the regular compatible splits and the compact radial components already established. Its status pieces are exactly the corresponding compact packet truncations. If n-1 walls met, assign those labels hit status and choose a closed sign status for each other processed label containing the point. The corresponding nonempty truncation would have dimension one, a contradiction. Forced signs persist under the continuous collapse recursion and enlargement of the cores. In particular

Properness of q_r and compactness of the original closed outer ball make each wall component compact; regular local wall charts give local finiteness, as in Proposition A.5.

The horizon identifications in Proposition A.5 preserve collapse, walls, and signed statuses. Its collar compressions preserve the old strata and their sides, and Remark A.6 allows their collapse oscillations to be bounded by the prescribed \eps_r. These choices are made separately on the locally finite compact component family; no uniform collar width or horizon at infinity is required. The next section chooses these finite-stage homeomorphisms coherently on one fixed copy of M.□

Proof. Choose positive numbers \eps_0,\eps_1,\ldots with \sum_{r\ge0}\eps_r<1. Here \eps_0 is assigned to the single radial stage; the other numbers are assigned to successive color stages. In this proof only, index the post-radial reconstruction by r=0. For a finite reconstruction write

The map q_r is the proper collapse of Proposition A.5, whereas H_r is a homeomorphism to a fixed copy of the original manifold. We maintain

Walls and their two closed sides will also be viewed on this fixed copy using H_r.

For the radial stage, the cut has a locally finite family of compact components. Choose pairwise disjoint, locally finite collars of these components on which the collapse varies by less than \eps_0 in each normal segment. Compactness of each component and continuity of the collapse permit the collars to be this small, independently from one component to another. Compress the finite inserted cylinders into these collars. Since their supports are disjoint, the displacement bound is \eps_0, rather than a sum over components. This constructs H_0 and proves (7.4) initially. The finite length of an inserted cylinder may be arbitrarily large; its length does not impose a lower bound on the width of the collar into which it is compressed.

We describe a subsequent color stage i. First raise any old finite horizons needed for the new compatible defining functions. Denote the resulting reconstruction by \widehat X_r. The horizon reparameterization

preserves collapse and every old wall and signed side. Indeed, on an old product end the collapse is independent of the normal coordinate, and an increasing change of that coordinate preserves its sign and its zero section. Compatible products of these changes agree at corners. Thus we may set

In particular, increasing a horizon contributes no displacement on the fixed copy.

The new wall lies over B_0(x_i,2D). Choose its collar strip S_i in \widehat X_r so that

and so that collapse oscillation along every segment used by the compression is less than the error assigned to this stage. These choices are possible because the relevant wall components in a finite reconstruction are compact and locally finite. More specifically, their union lies in the inverse image of the compact ball \overline B_0(x_i,2D) under the proper finite collapse; hence only finitely many components are involved in this color stage.

The strips can be chosen to preserve every old stratum, including the radial strata. To check this assertion, work near a point of a new wall on an old stratum, and denote the new defining function by u. The horizons were chosen beyond the product thresholds, so the terminal collars glued at each old hit wall lie in the product range. In the resulting product chart, u is pulled back from the corresponding face and is independent of the old signed normal coordinates. Its restriction to that face is regular at zero. In the corresponding product chart choose a vector field V in the face directions with V(u)=1, and with zero component in every old normal direction. The chart may avoid the old walls not passing through the point. A partition of unity combines these fields: the linear identity V(u)=1 and tangency to every old wall both survive. Its short flow supplies a collar preserving the old walls and their sides. On a fresh cylinder, each old stratum is the product of its intersection with the new wall and the cylinder interval. The compression uses the same increasing normal reparameterization on all these products. It consequently preserves every old signed side as well as all their intersections.

Let

be, respectively, the new cylinder collapse and the homeomorphism that compresses those cylinders into the chosen strips. They agree off the inverse image of S_i. Figure 2 shows the two maps schematically in a normal coordinate: c_r collapses the inserted interval, while h_r compresses it into the chosen old collar. Define

The oscillation choice gives d_0(Q_{r+1}(x),Q_r(x))<\eps_{r+1} wherever they differ, proving (7.4) at the next stage. Their difference is supported in \widehat H_r(S_i). If x=\widehat H_r(z) belongs to this set, then (7.5), (7.6), and (7.4) imply

This support radius is independent of the stage.

The balls in (7.8) form a locally finite family. Consequently, on every relatively compact neighborhood in the fixed copy, the maps Q_r are exactly unchanged after finitely many stages. Their locally stationary limit q is continuous and satisfies (7.1). It is proper: the inverse image of a compact set is a closed subset of its compact closed 1-neighborhood in (M,d_0).

At each finite stage the signed sides are the closed reconstructed sides, with their intersection equal to the central wall. The stratum-preserving identifications just constructed preserve every previously installed side and wall exactly. Newly installed color walls are confined to the fixed balls in (7.8). Thus all the partition data are locally eventually unchanged. Their final closed sets satisfy (7.2) and (7.3), since these are the finite-stage forced-sign properties of Proposition 6.6, and both collapse and statuses are stationary near each point. The wall family is locally finite by the same support bound. Finally, a point on n-1 distinct final color walls would, on a neighborhood where these data have stabilized, lie on the same n-1 finite-stage walls. This contradicts Proposition 6.6.□

Proof. Set O_i=q^{-1}(B_0(x_i,2D)) and F_i=q^{-1}(A_i\cup B_i). By Proposition 7.1, W_i is disjoint from the closed set F_i and lies in the open set O_i. At a point x, local finiteness and closedness of the walls give an open neighborhood meeting only walls that contain x. Shrink it to a neighborhood U_x contained in O_i\setminus F_i for every such wall. Therefore U_x meets at most k walls, and

Choose a locally finite open star refinement \mathcal V of this cover: the union of the members of \mathcal V meeting any fixed member is contained in some U_x. For completeness, such a refinement can be obtained directly with the metric d_0. The function

is positive and 1-Lipschitz, with distance to the empty set truncated to 1. The balls V_x=B_0(x,\rho(x)/8) star-refine the original cover. Indeed, if V_y meets V_x, the Lipschitz inequality gives d_0(x,y)<2\rho(x)/7, and hence V_y\subset B_0(x,25\rho(x)/56). The latter ball is contained in B_0(x,\rho(x)/2), which lies in a member of the original cover. Fix o\in M. Obtain a locally finite open refinement of (V_x) by covering each compact annulus \overline B_0(o,m)\setminus B_0(o,m-1) with finitely many open subsets of members of (V_x) lying in \{m-2<d_0(o,\cdot)<m+1\}, for m\ge1. These sets cover M and are locally finite. A refinement of a star refinement is still a star refinement of the original cover, which proves the assertion about \mathcal V.

Define

It contains W_i, and (7.10) implies the containment in (7.9). If z belongs to several T_i, choose V_0\in\mathcal V containing z. For each such i there is a V_i\in\mathcal V containing z and meeting W_i. All the V_i meet V_0, so their union lies in a single U_x. That U_x meets each of these walls, and therefore there are at most k of them. Finally, (T_i) is locally finite because T_i\subset O_i and the family (O_i) is locally finite.□

Proof. Use the cubes Q(S,A) and staircase simplices of (7.12), where S,A are disjoint finite sets, S\ne\varnothing, and |A|\le k. Realize the abstract simplices as standard simplices, and send each coordinate simplex to its new realization with the same barycentric coordinates. The rules agree on common faces. Equality of two images identifies their minimal simplex and its positive barycentric coordinates; the affine combination of the corresponding finite indicator vectors recovers the same original coordinate vector. Thus the conversion is injective.

For k\ge1, give each simplex edge length 1 and define d_K as the infimum of the total Euclidean lengths of finite piecewise straight paths in the simplices. This infimum is finite: vertices \mathbf1_S and \mathbf1_T are joined by adding the coordinates in T\setminus S and then removing those in S\setminus T, always retaining a nonempty set of coordinates equal to 1. Every point is joined to a vertex in its simplex. For a vertex v, its barycentric coordinate b_v, set to zero on simplices not containing v, agrees on faces and is \sqrt2-Lipschitz on each unit-edge simplex. Indeed such a simplex is realized by the vectors e_v/\sqrt2 in a Euclidean coordinate space. Hence

The barycentric coordinates distinguish points, so d_K is a genuine finite metric. For k=0 use the discrete vertex metric.

Near a point of Y, only a finite set F of coordinates can be nonzero. The coordinate map there lands continuously in the finite cubical subcomplex with all other coordinates zero. The conversion is continuous on each of its closed simplices and hence on the finite complex. Inclusion into the full uniform realization is continuous on a simplex because distance there is at most the length of its straight segment. Finite pasting therefore proves continuity on the entire finite subcomplex, and thus near every point of Y. The same local finite argument applies for k=0. No equality between two topologies on an arbitrary infinite complex is required.□

Proof. Choose an auxiliary smooth Riemannian metric on N to define gradients. The critical set is closed and its points are isolated. Properness therefore makes the number of critical points in any compact slab u^{-1}([a,b]) finite. Thus the critical values form a locally finite subset of \R, even if several critical points have the same value. Every level is compact and locally path connected: regular charts give a hypersurface, while the Morse normal form at a critical point gives the zero set of a nondegenerate quadratic form, which is a cone or an isolated point. A compact locally path connected space has finitely many components, since its components are open and form an open cover.

Let S consist of the critical values and the integers. It is locally finite. At each a\in S put a vertex for every component of u^{-1}(a). Between consecutive values a<b of S, the flow of \nabla u/|\nabla u|^2 identifies u^{-1}((a,b)) with a regular level times (a,b). The flow exists for each compact subinterval because its trajectory stays in a compact slab. Each of the finitely many connected product components defines an open edge, parametrized by its u-value. The image of a proper map to \R is closed; any finite endpoint of u(N) is an attained extremum and hence a critical value. Thus no further finite endpoints are missing.

An edge component E has a unique attachment at each end. At its lower end, the nonempty compact connected sets

are nested as \epsilon\downarrow0. Their intersection equals \overline E\cap u^{-1}(a): a point of this latter set is approached by points of E whose heights tend to a, and therefore belongs to every displayed closure. The intersection is nonempty, connected and contained in u^{-1}(a), hence in exactly one of that level’s components. Attach the edge to its vertex. The same argument applies at b. There are finitely many edges in either adjacent slab, so every vertex has finite valence. The resulting graph is countable and locally finite. Its points correspond exactly to the connected components of all levels, using full level components as vertices when critical values coincide.

Define q by this correspondence. On an open edge it is continuous by the product description. At a vertex corresponding to a compact level component C\subset u^{-1}(a), an adjacent edge not attached to C has no sequence approaching C: such a sequence would put a point of C into its endpoint intersection, contrary to its unique attachment. The finitely many other components of u^{-1}(a) are disjoint compact sets. A neighborhood of C can therefore exclude them and the finitely many nonincident edges. Requiring also that |u-a| be sufficiently small places its remaining points in any prescribed short tails of the finitely many incident edges. This proves continuity at the vertex and shows that its whole inverse image is C. The graph is connected, since q is a continuous surjection and N is connected.

Subdivide every edge once with a different midpoint vertex for each edge. This removes parallel edges from the abstract realization; loops do not occur because the endpoint heights differ. The result is a simplicial graph. Give each new edge the same length. Finite valence implies that its ordinary graph topology equals the topology of this uniform simplex metric: vertex neighborhoods contain short segments on each of finitely many incident edges. The subdivision and change of edge lengths preserve points and fibers. Thus q has the asserted continuity and exact fibers.□

Proof of Corollary 1.3. Put s=\min_M\Scal_h>0 and X=\widetilde M. The lifted metric is complete. If X is compact, use a constant map to a vertex, whose only nonempty fiber has finite diameter.

For n\ge4, the complete metric s\widetilde h has scalar curvature at least 1. Apply Theorem 1.1 using the coordinates of (7.11), then Lemma 8.1. Local finiteness of their supports was proved in Section 7, and the conversion preserves their entire fibers. Scaling back gives the bound C_n/\sqrt{s} in the original metric.

For n=2, the Gaussian curvature of the cover is at least s/2. If a unit-speed minimizing geodesic had length L>\pi\sqrt{2/s}, take a parallel unit normal E along it and the endpoint-zero variation field V(t)=\sin(\pi t/L)E(t). Its index form obeys

contradicting minimality. Completeness supplies minimizing geodesics, so the cover has diameter at most \pi\sqrt{2/s} and is compact by Hopf–Rinow. The constant map therefore proves this case.

Finally let n=3 and X be noncompact. Apply Liokumovich and Wang [26, Theorem 1.1, arXiv version 1] to the complete metric (s/6)\widetilde h, whose scalar curvature is at least 6. It gives a proper Morse function such that every connected component of every level, including critical levels, has ambient diameter less than 18\pi in that metric. Apply Lemma 8.2. Each whole fiber of its graph map is one such level component, so in the original lifted metric it has diameter less than 18\pi\sqrt{6/s}. This is a continuous map to a one-dimensional uniform simplicial complex, as required.□

Proof. First work in the finite downward diagram of one old packet. Raise all entry thresholds until every overlap is in a commuting quadrant, and until all the functions in this diagram are products there. For an elementary normal coordinate t_i choose a larger horizon h_i. On its tail use t_i\mapsto\min(t_i,h_i) and the homotopy t_i\mapsto(1-s)t_i+s\min(t_i,h_i); use the identity elsewhere. The formulas agree continuously at t_i=h_i. In an overlap the other normal coordinates are unchanged and the formulas commute. One may therefore perform these clamps in any order. Their joint image is compact: remove the far elementary tails, and then inductively truncate each of the finitely many face tails. This gives a finite union of compact products. Its boundary strata have precisely the claimed product descriptions. Each clamp fixes the cutting function, since that function is a face pullback throughout the moving region.

The regular zero set in this compact manifold with corners is transverse to every corner stratum. Indeed on a far old face the derivative in face directions is the derivative of the regular face function. Consequently the compact zero set and the two compact closed sides are manifolds with corners. Each has finitely many components: every point has a connected neighborhood in the relevant manifold with corners, and compactness supplies a finite such cover. Clamping shows that a full zero or side component has a nonempty connected intersection with the truncation. Conversely a connection in the full set clamps to a connection in the truncation. Thus no additional components or connections occur only at infinity.

On an old tail the whole operation is literally the same operation on its face times the old normal intervals. In particular the cylinder attached to a face zero set is the same subset in every order of passing to that face. For the new sign-to-hit relation the tail is exactly the newly attached cylinder. These observations prove all product and commutation rules, including the unique occurrence for a prescribed coarser status: the outgoing product of a connected face packet is connected and belongs to a unique coarser component. A new packet has finitely many faces by the compact truncation just constructed. A face has at most one incident packet for each coarser status, of which there are finitely many. Applying these assertions through the finite status depth proves finiteness in both directions.

For clarity, properness of tails is part of this argument, not an inference from a metric completeness assertion. Fresh closed cylinder tails are proper by the attachment topology. The continuous collapse of a new space to its old parent sends a compact set to a compact set. On an inherited tail, old properness then bounds its old parameters in a compact subset of the old tail; the fresh cylinder length is also bounded on the given compact set. The remaining zero and side conditions are closed. Thus the inverse image of that compact set in the inherited closed tail is compact. Equivalently, it is contained in the union of closed old pieces and bounded fresh cylinders over compact zero pieces. This also shows that the complement of all new far tails is compact. Clamp the fresh cylinder coordinate as well to obtain the asserted new compact truncations and retractions.

At the radial stage there are initially no old tails. The hypotheses give compact wall and side pieces, with finite incidence on each compact piece, and the fresh cylinders start the preceding induction. All diagrams remain countable, because all manifolds involved are second countable and the radial family is locally finite.

For later metric uses, normal collars can be obtained from any compatible smooth parent metric. In an old product tail, both the regular defining function and the metric are pulled back from the face, so the unit normal and its normal geodesics split off the old normal coordinates. On a compact truncation of the zero set choose a positive tubular radius by finitely many normal-coordinate charts. Induction through its finite face diagram extends this radius to a positive uniform radius on that parent packet: any failure would be in a product over a face where the radius was already chosen. Thus these collars are products on all old ends; at the radial step the same choice is made separately on each incident compact block.□

Proof. Induct upward from packets with no faces, which are compact. A full elementary product tail over a complete face has no finite-distance exit through its face coordinate. More precisely, a finite-length curve in that product whose normal coordinate stays above the entry level has Cauchy face and normal coordinates. Completeness of the face and properness of the closed tail keep its limit in the product. It follows that, after fixing a closed-tail entry level T_i, the function

is continuous and globally 1-Lipschitz. The only exit is through t_i=T_i, and its differential has norm at most one wherever defined. A Cauchy sequence has all the finitely many \rho_i bounded. The compact-truncation assertion of Lemma A.1 puts such a sequence in a compact set. Hence the packet is complete.

Recursively subdividing each coordinate into a bounded interval and its far range gives a finite cover of E by compact sets and sectors

where K_B is a compact subset of a further face and the metric and warps are products. The recursion terminates because a face decreases base dimension. On such a sector each displayed normal coordinate is also the corresponding elementary coordinate on E, by commutation. In an R-ball its range therefore has length at most 2R, by the Lipschitz functions \rho_i (enlarge by a fixed bounded interval at an entry if necessary). The sector’s weighted volume in that ball is bounded by its compact base volume times a constant multiple of (1+R)^d. Summing the finitely many bounds proves Equation (A.1), for all centers, including centers outside the sector. Notice that d\le\dim E.

Bounds on smooth data follow from the same finite sector decomposition: on compact pieces they are finite, and on a product they are precisely face bounds. For bounded geometry one can equivalently use uniform charts. Any failure of a uniform chart radius or of a fixed-order chart bound has an escaping sequence; a subsequence lies in a far product over a compact part of a further face, as detailed in Lemma A.3. Fixed-radius neighborhoods there are product neighborhoods with uniform smooth chart bounds and a positive injectivity radius. Compact remaining regions have the same properties by a finite chart cover. This proves bounded geometry and all warp bounds. The product assertions follow either by the same proof or from B_{E\times\R^a}((x,z),R)\subset B_E(x,R)\times B_{\R^a}(z,R).□

Proof. Choose a sector from the finite cover in Equation (A.2) containing a subsequence. Pass to a subsequence so each of its normal coordinates either converges finitely or tends to infinity. The compact base coordinate also converges. Absorb the bounded normal coordinates into the base of the intermediate face specified by the remaining, unbounded coordinates. Commutation identifies this with the corresponding full product occurrence. At least one unbounded coordinate remains, since otherwise the sequence lies in a compact truncation. A path of length at most R changes each normal coordinate by at most R. It cannot escape through a face coordinate, by completeness of the face as in Lemma A.2. Thus a fixed-radius neighborhood lies in the product once all its normal coordinates exceed the entry levels by more than R. Translating yields the stated exhaustion. This argument uses only the completeness part of Lemma A.2, which was established before its bounded-geometry conclusion.□

Proof. There are only finitely many requests on a particular face, because there are only finitely many coface chains. Starting with the given requests, propagate stricter tolerances down these chains, allowing for the finitely many product, derivative, and cutoff constants. Take finite minima; no minimum over an infinite adjacency component is taken. This finishes all tolerance choices before construction begins. A countable collection of packets causes no problem, since each one has finitely many faces and cofaces and the depth is finite.

For the construction on one packet, enumerate its elementary tails. Choose smooth cutoffs f_i(t_i) rising from zero to one sufficiently far out, with bounded derivatives through the chosen order, and extend them by zero off their tails. Write u_0 for an individual approximation and u_i for the already chosen face copy. The weights

produce w_0u_0+\sum_iw_iu_i. Put the transitions beyond all relevant intersection thresholds. On an overlap the participating face copies then agree by their own product rules. In particular, wherever some f_i=1, every nonzero term is the same face copy. The result is an exact copy there. Leibniz’ rule bounds the derivative error by a finite multiple of the preassigned errors. A perturbation on the remaining compact region changes none of these end identities. This proves the conditional approximation assertion. It does not assert convergence for a temporary nonproduct object; whenever such an object is used below, its convergence is proved separately.

For positive small functions start with positive constants on the compact deepest packets, take positive small individual choices on compact parts, and use the positive weights above. Decrease the preassigned tolerances to cover all coface derivative requests. This gives strict positivity at every point and any stipulated finite norm bounds, including the C^2 bounds used for seam widths. For the separation assertion, a sequence of distances tending to zero either stays in a compact region, where closedness and inclusion in the open set give a contradiction, or escapes. In the latter case Lemma A.3 supplies a common product window over a compact part of a deeper face. The two sequences project to the same limit point, lying both in F_B and in B\setminus O_B, a contradiction.□

Proof. We prove the assertions in their order of dependence. At stage zero use X_0=X and q_0=H_0=\Id. At the radial split, the assumed compact pieces give the first finite reconstruction by inserting cylinders of finite length on the two sides of each wall component. The properness and homeomorphism argument below applies to this locally finite compact family. Its compact pieces also initialize the analytic diagram by Lemma A.1.

Suppose the reconstruction through stage r has been made. Fix all product thresholds for the next cutting functions and their collar metrics in the full analytic coordinates first. Consider a connected component W of an existing global wall. A packet with the corresponding hit status has a nonempty connected truncation, by the commuting retractions of Lemma A.1; it therefore lies in a single such W. The same assignment holds at all incidences through further faces, since the corresponding truncations glue along those faces. Compactness of W and local finiteness of the reconstructed pieces imply that only finitely many such pieces meet W. Thus there are only finitely many threshold requests for its normal coordinate. Choose one horizon exceeding all of them, with a positive buffer. Do this for each existing wall component and coordinate.

This is a direct choice from previously fixed finite lists. Increasing a horizon cannot introduce a new analytic threshold: the functions and the incidence diagram were fixed on the full infinite tails. On a tail where topology and q_r are already products, choose an increasing smooth reparameterization from the enlarged finite normal interval to the old one, equal to the identity before the old product range and sending the new endpoint to the old endpoint. Choose the same reparameterization at all occurrences of W, on both sides of its central hit wall. It preserves those sides and the hit wall. At corners the separately labelled normal reparameterizations commute. Their product gives an identification \Phi_r:\widehat X_r\to X_r preserving every wall and sign status. The glued position maps satisfy

because each position map is independent of the moved normal variables. Properness is preserved by this homeomorphism. Thus horizon enlargement introduces no error or change of position data. For the rest of the step, relabel these enlarged representatives as X_r,q_r,H_r.

The compatible cutting functions now glue to a single smooth function on this enlarged X_r. Near any old wall stratum the function is a pullback from the hit piece, independent of all its signed normal coordinates. Its zero set L is therefore regular and transverse to every old wall stratum. More explicitly, in a chart (b,s_1,\ldots,s_d) at an old stratum, the old walls are s_j=0 and the new function is g_B(b). The face hypothesis says \dd g_B\ne0 at its zeros. Hence the old normal covectors \dd s_1,\ldots,\dd s_d together with \dd g_B are independent. This also covers a stratum involving the radial wall. At this point L has been formed without choosing a new horizon. For a color split its forced signs give

In the present one-ball-per-label setup, L\subset q_r^{-1}(B_{d_0}(x_i,2D)). The latter set has compact closure because q_r is proper and the original closed outer ball is compact. Since L is closed in X_r, the whole color wall L is compact. In particular each component is compact, and there are only finitely many components. More generally, a closed regular hypersurface with compact components has a locally finite component family: a regular hypersurface chart meets only its own component, and compact subsets of the hypersurface have finite such covers. The first radial wall need not be compact as a whole; its stated locally finite compact-component hypothesis gives exactly what is needed for the same construction.

Only now assign a finite new horizon to each component of L. Cut along L, attach a cylinder on each side, and identify the two outer cylinder boundaries as the new central hit wall. Near an old wall this is the product of the operation on its hit face with the old signed normal coordinate. The same statement holds at multiple old intersections. Thus old walls extend smoothly through the new cylinders, and all walls have transverse intersection charts. The resulting pieces are exactly the truncations of the newly appended statuses: the old clamps preserve L and its signs and extend across fresh cylinders keeping their new coordinate fixed; the fresh-coordinate clamp commutes with all of them. In particular every full new packet retracts onto one nonempty connected piece in the reconstruction. This verifies the component assertion needed to assign its coordinate at the next stage.

Let c:X_{r+1}\to X_r collapse the inserted cylinders to L and be the identity on the bulk. This map is proper. A compact subset of X_r meets only finitely many new wall components; its inverse image is contained in its compact bulk copy together with finitely many finite cylinders over compact subsets of these compact components. The glued map is q_{r+1}=q_r\circ c, and so is proper. Every old compact wall meets finitely many new components, and its transverse intersection with them is compact. Its extension adds only finitely many compact finite cylinders; the old wall therefore remains compact. Local finiteness of all pieces follows locally from the finite collection of transverse wall charts, together with the compact truncation descriptions. This proves (i) and the properness assertion without using any later wall.

We next choose collars that preserve all old strata. The compatible packet metrics glue on the enlarged reconstruction: near an old hit stratum they have the form h_B+\sum_j\dd s_j^2. On a neighborhood of L, write u for the glued cutting function and use the transverse field

In each old product chart, u=g_B(b) and this field has no old normal component. Thus it is tangent to every old wall meeting that chart and is the pullback of the corresponding face field. Equivalently, one can patch such local fields with a compatible partition of unity: V(u)=1 and tangency are linear conditions, so they survive patching. The flow supplies product collars at old intersections. It preserves every old wall and each side, by uniqueness of the flow and continuity from the identity. This explains why a transverse ordinary collar chosen without the product condition would not suffice here.

Choose pairwise disjoint locally finite strips around the compact components of L using this flow. A positive strip width exists on each component by compactness; it may be decreased to keep strips disjoint. For each component, continuity of q_r and compactness permit the width to be so small that q_r-images of any two points on the same flow segment have d_0-distance less than one. A monotone normal reparameterization compresses the two finite inserted intervals and the old strip into that old strip, fixing its outer boundary. It produces a homeomorphism h:X_{r+1}\to X_r, equal to the identity off the strips, with

Different strip widths and cylinder lengths are allowed on different components. Local finiteness makes these definitions a global homeomorphism. Set H_{r+1}=H_r\circ h. The triangle inequality and Equation (A.4) give the new bound r+1. The compression follows the stratum-preserving collar, so it also preserves every old wall and its two side statuses. The old signs are pulled back under c, including across the cylinders, and the newly imposed signs hold on the prescribed inverse images. Pullbacks of their open phases stay open. This proves (iv) and finishes the induction.□

Proof of Lemma 5.4. The decreasing solution of 2|y'|=.5y^2+\pi_0, from +\infty to -\infty, has finite crossing length

Extend it by +\infty before the crossing and by -\infty after it. This is continuous as an extended-real-valued function, and all its finite truncations are locally Lipschitz. Composing the transition, begun at distance 2 from A_i, with the 1-Lipschitz function d_h(A_i,\cdot) gives

almost everywhere on the finite band. Take

Their closures lie strictly inside the respective pole interiors; they contain A_i and B_i. The requested finite separations are available from Theorem 4.3 after N_*, and hence T_0, are fixed. Their size does not change the original radius D.

We give the radial construction without a bounded-geometry hypothesis. Assume first that X is noncompact, fix o\in X, and put d(x)=d_h(o,x). Choose a countable locally finite cover by relatively compact coordinate neighborhoods on which the metric oscillation is small, and a subordinate smooth partition of unity \{\chi_j\}_{j\ge1} whose supports are compact in those neighborhoods. Convolution in a slightly larger coordinate neighborhood gives smooth d_j with

The derivative estimate follows from the local Lipschitz bound for d and the small metric oscillation; the convolution radius can then be reduced to make its value error arbitrarily small. Let C_j=\sup|\dd\chi_j|_h<\infty, and choose 0<\epsilon_j<\min\{1/2,2^{-j-2}/(1+C_j)\}. The locally finite sum \rho=2+\sum_j\chi_jd_j is smooth and |\rho-(2+d)|<1/2. Moreover, since \sum_j\dd\chi_j=0,

Completeness makes d proper, so \rho is proper and bounded below. Thus a smooth proper radius with bounded gradient has been constructed using only local smoothness and completeness.

Choose successive plateau intervals of the values of \rho, with alternating signs and positive-width cores strictly inside each plateau. Separate successive plateaus by transition intervals of width greater than 2T_0. On such an interval, a copy of the crossing y, parametrized by (\rho-a)/L with L>2 and with sign reversed on alternate intervals, obeys the same profile inequality because |\dd\rho|_h/L\le1. Include an initial plateau extending below the minimum of \rho. Core closures have positive value margins to both ends of their plateaus, except at the harmless lower end of the initial plateau. Sard’s theorem supplies regular values in those margins. The sublevel domain at the initial plateau, and the domains between two such regular levels at later plateaus, are compact smooth barriers enclosing the corresponding core closures and lying in the same pole interiors. These barriers and the transition blocks form locally finite families, since their value intervals tend to infinity. Connectedness and properness imply that the image of \rho contains all sufficiently large values, so the separating subbands are nonempty.

A zero wall respecting the signs misses every separating core subband. On any connected wall component, the continuous function \rho therefore takes values in one bounded intervening block. The component is closed in X, and properness of \rho makes it compact. For a fixed sign, successive core subbands of the opposite sign bound the \rho-values of every connected component of the closed side; the central component is bounded by the first opposite separator. These closed-side components are compact as well. In a compact block, regularity gives a finite cover by charts in which the wall is a single sheet and each closed side is a half-space or the full chart. Only finitely many components can meet that block. The blocks are locally finite, which proves the required local finiteness of wall and side components.

If X is compact, set the radial profile identically +\infty, with U_{\mathrm{rad}}^+=X and U_{\mathrm{rad}}^-=\varnothing. The constant positive defining function realizes its empty wall; the single nonempty closed side is X, which is compact. All radial profile and confinement assertions then hold directly.

Finally, the initial scalar requirement holds everywhere, since

The profile, core, and forced-preimage conditions have already been verified. This proves the invariant at stage zero.□

Proof. The nesting of the core closures inside the pole interiors and the compatible approximation construction of Lemma A.4 give smooth separating cutoff functions. A level which is regular on every packet gives the required domains: there are only countably many packets, so Sard’s theorem permits a common level. Their far collars are exact copies of the face domains. On a fixed packet, compact parts and the finite face diagram give a positive tubular radius and bounded geometry for the resulting boundary. Extend its outward normal in a smaller tube and multiply by a cutoff of absolute value at most one. This supplies Y^\pm and a bound for its weighted divergence. The radial case uses the relatively compact pole barriers individually.

Fix the barriers and their vector fields first. By Lemma A.4, choose a compatible positive smooth function h with sufficiently small value and first derivative, and put M=h^{-1}. Require

Only finitely many coface requests reach a given packet; at the radial step the requests are locally finite on compact sets. The choices therefore have the stated compatibility. The derivative inequality implies 2|\grad M|\le a_rM^2. The clipped function

is finite and locally Lipschitz. At almost every point its derivative is that of one active branch, and consequently

This assertion also holds near the original poles, since the clipping levels are locally bounded and finite truncations of \mu_{j_0} are locally Lipschitz. On D^\pm one has z=\pm M.

Here is the quantitative smoothing fact being used. If a locally Lipschitz function z obeys |\grad z|\le F almost everywhere for a continuous positive F, then, for arbitrary positive continuous error targets, local convolution in sufficiently small metric charts followed by a partition of unity gives a smooth z_1 with prescribed value error and |\grad z_1|\le F plus the prescribed error. Indeed convolution averages the gradient inequality, the oscillation of F and the metric on its support tends to zero, and differentiating the partition of unity produces only the value errors times its fixed derivatives. A locally finite cover permits the errors to be assigned summably at each point. For compatible smoothing, start on deepest faces. On overlaps in far collars the clipped functions and all face copies agree. Blend with the cutoffs in Lemma A.4; the same value-error estimate pays the additional cutoff derivatives. Preallocate smaller errors on faces to meet their finitely many coface requests. Thus the gradient estimate is always compared with the single continuous function (a_rz^2+\Pi_{j_0})/2, rather than with unrelated bounds for the pieces.

Choose the value error at most \min\{1/4,10^{-3}\sqrt{\Pi_{j_0}}\} and the gradient error so that

These are positive continuous error targets. For e=p_0-z,

so a_r\le .63 gives

On D^\pm the value error and the earlier choice of M give \pm p_0\ge 11/4+|\Div_\omega Y^\pm|. Finally set

The elementary identity

and \Pi_{j_0}\ge\pi_{\rm cut} prove (B.1) for every |c|<c_*; the barrier margin also persists. Thus the domains, clipping height, smoothing, and open shift interval have all been fixed before selecting a minimizing phase. On a post-radial packet every smooth datum just constructed has bounded derivatives, by its compact parts and exact products. At the radial step the construction gives the corresponding bounds on each compact window, which is all that is used.□

Proof. We first establish the comparison formulas that will show both minimality of a limiting phase and absence of perimeter loss. We then construct exhaustion minimizers and apply these formulas to locally convergent sequences.

Comparison formulas. Local L^1 compactness under local perimeter bounds and lower semicontinuity are Simon [36, Chapter 2, Theorem 2.6]; positive smooth weights and smooth metrics on a relatively compact chart give equivalent variation norms. The further BV inputs are the perimeter lattice inequality and the trace and gluing formulas, both across a fixed smooth face and across almost every admissible level of a Lipschitz slicing function. The lattice inequality is

For smooth functions u,v, the gradients of \min(u,v) and \max(u,v) are the two original gradients, possibly interchanged, almost everywhere. Strictly approximate the two characteristic functions, apply this identity, and use lower semicontinuity on the minimum and maximum. This gives (B.4), with its smooth metric and weight.

We record the trace argument for a fixed smooth comparison face, since such a face need not be an almost-everywhere selected level. Work first in a flattened coordinate collar Q\times(-a,a)\subset\mathbb R^{m-1}\times\mathbb R, with the face T=Q\times\{0\} and the plus side given by t>0. All collars used here are buffered inside a relatively compact ambient chart. The functions under consideration are integrable and have finite total variation on each smaller collar reaching T; for the restrictions of our finite-perimeter phases this follows from local finite perimeter in the ambient chart. Local BV regularity only on sets bounded away from T would not suffice.

Let w be such a BV function on the plus half-collar, and let D_t w denote its distributional derivative in the t direction. Choose a compact tangential box K and an open box K' with K\Subset K'\Subset Q, and fix a_0<a. Mollify on the open half-collar, using kernels small enough to remain inside it on each fixed strip. Fix a countable exhaustion by buffered tangential boxes and compact height intervals contained in (0,a). Choose one subsequence whose local L^1 errors are summable on every member of this exhaustion, by making its kth error at most 2^{-k} on each of the first k members. Fubini then gives convergence in L^1 on horizontal slices for all heights in one common set of full measure, simultaneously on the exhaustion boxes. Restriction gives the same convergence on K'. For a smooth mollification the fundamental theorem of calculus, followed by integration over K, bounds the difference of two slices by the integral of the absolute vertical derivative on the intervening strip. Convolution bounds this integral by the variation of D_t w on a slightly enlarged strip; the tangential enlargement lies in K' when the kernel is small enough. In the limit the enlarged tangential box lies in K' and the height interval is contained in [s,t]; thus the limiting bound is at most |D_t w|(K'\times[s,t]). Excluding heights at which |D_t w|(K'\times\{t\})>0, gives

for all s,t in a common set of full measure. There are at most countably many excluded atom heights, since the variation on the buffered collar is finite.

The right-hand side of (B.5) tends to zero uniformly for good heights s,t\downarrow0: the sets K'\times(0,\delta) decrease to the empty set inside the open half-collar and have finite variation measure. The good slices are therefore Cauchy in L^1(K) and have a limit, denoted \operatorname{Tr}_+w. Traces obtained on nested tangential boxes agree by uniqueness of the L^1 limit, giving a locally integrable trace on T. Reflection gives the analogous trace \operatorname{Tr}_-w from the minus side. For characteristic functions these traces are themselves characteristic functions almost everywhere, since an L^1 limit of characteristic functions has an almost-everywhere convergent subsequence. In particular the phase traces take values in \{0,1\}.

The trace also gives the required integration-by-parts formula. On the plus side, multiply a smooth vector field, compactly supported in the tangential directions and away from the outer end of the collar, by a cutoff which is zero for t\le\varepsilon and one for t\ge2\varepsilon. Apply the distributional derivative identity inside the open half-collar and let \varepsilon\downarrow0. The cutoff derivative term tends to the pairing of \operatorname{Tr}_+w with the vector field’s t component at T: its derivative has integral one, and the good slices converge in L^1 uniformly as their heights approach zero. The other terms converge by integrability and finite variation. This gives the boundary term with outward normal -\partial_t on the plus side; on the minus side the outward normal is \partial_t. Undoing the smooth coordinate change gives the corresponding Riemannian formula. More explicitly, the smooth coordinate volume density is included in the divergence test; its boundary flux is exactly \langle X,\nu\rangle\,\dd\mathcal H_g^{m-1}. Thus, if D_g w denotes the intrinsic distributional gradient measure, with convention \int w\Div_g X\,\dd\vol_g=-\int\langle X,\dd D_g w\rangle for tests supported in the open side, then

The vector field here has compact support in the collar, so no other boundary contribution occurs.

Now orient \nu_T from the minus side to the plus side, and glue a BV function u on the minus side to a BV function v on the plus side. Adding their integration-by-parts identities shows that the gradient measure of the glued function is

These three vector measures are concentrated on the disjoint measurable sets U_-, U_+, and T. Their variations therefore add. Multiplying the variation measures by the smooth positive weight \omega gives the weighted formula, whose extra interface cost is precisely

We are weighting the variation, not differentiating \omega times the glued function. For phases the last integrand is the indicator of the trace-disagreement set, multiplied by \omega.

We will also glue across almost every level of a locally Lipschitz slicing function f, whose level sets need not be smooth. For this distinct use, the reduced-boundary representation identifies a phase locally with its integer-multiplicity integration current and identifies boundary mass with perimeter [36, Chapter 3, Section 4, Equation (4.1) and Theorem 4.3]. In a locally oriented chart the phase current has locally finite mass and locally finite boundary mass. Lipschitz coarea and the current slicing and restriction formulas therefore apply [36, Chapter 3, Section 2, Equations (2.8)–(2.9); Chapter 6, Section 4, Lemmas 4.1 and 4.5 and Definition 4.4]. In particular, writing J for either phase current, for almost every level \tau one has

Apply this identity to both phases, excluding also the null set of levels charged by their local volume or perimeter measures. For a top-dimensional phase current the slice has multiplicity one on the sliced phase and zero elsewhere. Indeed the slicing definition restricts its phase indicator to the noncritical part of the level; coarea shows that critical points and any changes of the ambient almost-everywhere representative affect almost no level in positive hypersurface measure. The two phase slices have the same orientation induced by f in the locally oriented chart, so their difference has multiplicity equal to the difference of the two sliced indicators. The boundary of their gluing has the two restricted boundary currents and the difference of their level slices as its interface term. Its extra weighted mass is bounded by the weighted hypersurface measure of the set where the sliced phase values differ. This is the almost-everywhere interface-cost bound used in the annular comparisons, distance-ball comparisons, and level-set deletions below. A countable collection of phases, including a converging sequence and its limit, can use one common set of admissible levels by intersecting their full-measure sets. If the phase indicators converge locally in L^1, coarea on a fixed buffered slicing annulus makes the integrated sliced disagreement tend to zero. Pass to a subsequence for which these integrals are summable. Fubini then makes the disagreement tend to zero at almost every level, still within the common admissible set. This supplies a fixed gluing level with vanishing interface cost in the compactness argument below. All orientations in this argument are local; global orientability is not required.

Exhaustion minimizers and local compactness. Choose a smooth relatively compact exhaustion U_k\Subset E. Minimize over phases vanishing outside \overline U_k, taking the total perimeter, including the interface on \partial U_k, and the pressure integral inside U_k. The empty phase is admissible. Because the pressure is bounded on \overline U_k, an energy bound gives a perimeter bound. The class is closed under L^1 convergence, the volume term is continuous, and perimeter is lower semicontinuous. The direct method thus gives a minimizer, locally absolute minimizing in U_k.

The same local bound applies to arbitrary local absolute minimizers. In a coordinate ball B_t compactly contained in their comparison domain, comparison with the phase cleared inside B_t gives, for almost every t,

Choose t in a fixed annulus around the smaller window to be bounded. Coarea bounds the first term for a suitable slice. Smoothly convergent metrics and positive weights and locally uniformly bounded pressures make this a uniform local bound. Diagonal BV compactness therefore produces a locally L^1 convergent subsequence, both for the exhaustion minimizers and for any sequence in the statement.

Minimality and perimeter convergence. The gluing formulas now apply to the locally convergent subsequence. Choose nested relatively compact windows U\Subset V with a Lipschitz foliation of the intervening annulus. The integral over slicing parameters of the trace discrepancy between A_i and A tends to zero by L^1 convergence and coarea. One can therefore choose a slicing hypersurface between U and V whose discrepancy tends to zero along a subsequence. Glue A inside this surface to A_i outside. Minimality of A_i bounds its interior perimeter from above by that of A, the vanishing trace cost, and a vanishing difference of volume terms. The last difference vanishes by the convergence of the data and indicators. First use windows whose limiting perimeter gives zero mass to their boundaries; then shrink the annulus. Together with lower semicontinuity this proves weak local convergence of the perimeter measures. Replacing the interior copy of A by any compact competitor B gives the same upper comparison with B, since B=A on the gluing annulus. Lower semicontinuity on the left proves (B.3) for A. Smooth metric and weight changes make only errors tending uniformly to zero on each fixed window, so the same argument proves the convergent-data assertion.

Changing metrics and varifold convergence. For the varifold assertion, account explicitly for the changing metric. In a coordinate chart let D_0\chi_{A_i} be the Euclidean distributional derivative and let (g_i) denote the positive metric matrices. The vector measures

have Euclidean total variation equal to the g_i-perimeter measures. The matrices and their derivatives converge smoothly, so distributional convergence and the local mass bounds give \mathbf v_i\rightharpoonup \mathbf v. The perimeter conclusion gives |\mathbf v_i|\rightharpoonup|\mathbf v| as well.

This strict convergence also controls the unit polar vectors. To see this, on a compact window approximate the limiting polar vector in L^2(|\mathbf v|) by a continuous bounded vector field e. Expand

Weak convergence gives its limit, and then the approximation makes that limit arbitrarily small. Uniform continuity consequently gives convergence for every continuous compactly supported integrand of position and polar vector. The unoriented g_i-orthogonal tangent hyperplane is a continuous function of this vector and g_i. This proves varifold convergence. Multiplication or division by the smooth positive weight on a fixed compact window gives the asserted weighted and unweighted versions.□

Proof. We first prove the positive assertion. Put D=D^+, Y=Y^+ and G=D\setminus A. For almost every distance radius R about a fixed point, U=B_R is a finite-perimeter set, its boundary has zero mass for the relevant perimeter measures, and BV traces are available. Write

Distance coarea, with |\grad d|=1 almost everywhere off the center, gives local absolute continuity and V'(R)=S_R for almost every R. Weighted Gauss–Green, Y=\nu_D on \partial D, and |Y|\le1 give

By (B.4) this bounds the interior perimeter increase on replacing A by A\cup D in U. Glue back to A outside U; the extra perimeter is at most S_R. This is a compact comparison in a slightly larger relatively compact open set. By (B.2) and minimality,

The weighted ball volume is at most polynomial by Lemma A.2, since \omega is bounded on the packet. If V(R_0)>0, (B.6) would imply V(R)\ge V(R_0)e^{(R-R_0)/2}, a contradiction. Thus G is null. For a relatively compact radial barrier, simply add all of D; no slicing terms occur, and the same comparison gives \mu(G)=0 directly. The complement of A minimizes pressure -p, proving the negative assertion. The domains contain the core closures in their interiors; the packetwise positive separation follows from exact products and the compact-part property. A phase equal to one or zero on an open set has no perimeter support there.□

Proof. On any one such space, consider arbitrary minimizers A for p_0+c_1 and B for p_0+c_2, with c_1<c_2. Put G=A\setminus B, \delta=c_2-c_1, and use the same distance-ball slicing as in the preceding proof. Replace A by A\cap B inside the ball and B by A\cup B there, gluing each to its own exterior phase. The two gluing costs total at most 2S_R. Their interior perimeter increments have nonpositive sum by (B.4); their pressure increments sum to precisely -\delta\mu(G\cap B_R). Adding the two minimality inequalities gives

Polynomial weighted growth, also on E\times\R^d, forces V=0. Thus every minimizer at the smaller shift is contained in every minimizer at the larger shift.

Choose countably many nonnegative compactly supported continuous test functions \varphi_a which separate locally integrable functions. For each a and shift c, let

Existence is supplied by Lemma B.3. The preceding ordering says that the nonempty open intervals I_a(c) for distinct c are disjoint. Each contains a rational number, so only countably many can be nonempty. If minimizers differ, some \varphi_a distinguishes them and I_a(c) is nonempty. There are therefore at most countably many exceptional shifts on this space. The packet collection and the set of integers d are countable, so their exceptional shifts have a countable union. Choose c in the remaining part of (-c_*,c_*).

A Euclidean translation of the unique minimizer on E\times\R^d is again a minimizer. Uniqueness implies translation invariance of its indicator. It suffices to impose the countable dense group of rational translations: continuity of translations in L^1_{\rm loc} then gives all translations, so its Euclidean distributional derivatives vanish. The indicator is therefore \chi_F(x) for a base phase F. BV slicing shows that F has locally finite perimeter. If a compact base competitor improved its energy by \eta>0, use that competitor over (-L,L)^d and the phase F outside, retaining the original phase outside the compact base discrepancy. The energy gain is \eta(2L)^d. The interfaces at the Euclidean box boundary cost at most O(L^{d-1}) times the weighted volume of that compact discrepancy. For large L this contradicts minimality. Thus F minimizes on E, and base uniqueness identifies it with A_E. The case d=0 needs no product argument.□

Proof by first variation, strict blowups, and regularity. Put d=m-1. The classical inputs are the bounded-mean-curvature monotonicity theorem and Allard’s graphical theorem [36, Chapter 4, Theorem 3.17; Chapter 5, hypotheses 5.1 and Theorem 5.2], and the singular-dimension theorem for exactly minimizing boundary currents [36, Chapter 7, Theorem 5.8]. The minimizing-boundary dimension bound follows from the work of Simons and Federer [14, 38]. The last theorem will be applied only to constant-metric blowup limits. We verify the transfer to the present smooth weighted problem, including the singular-set compactness needed for the Hausdorff-dimension conclusion.

First variation and local area bounds. First variation is available before any regularity conclusion. The flow of a compactly supported smooth field Z is an admissible comparison in (B.3), so

Here tangent planes and \nu are the measure-theoretic ones. Substituting Z=W/\omega cancels the tangential weight derivatives and gives

Thus the unweighted perimeter varifold has intrinsic generalized mean-curvature vector -(p-\partial_\nu\log\omega)\nu, locally bounded by |p|+|\grad\log\omega|. Its multiplicity is one almost everywhere.

On a relatively compact coordinate patch, first pass to a smaller patch and extend its metric to a closed manifold. Explicitly, place its coordinate closure inside a coordinate ball of a flat torus and blend the given positive definite metric with the flat metric outside the smaller patch; the convex combination stays positive definite. The resulting metric is smooth and the ambient manifold is compact, so the smooth isometric embedding theorem applies [33, Theorem 2]. The preceding intrinsic first variation then gives Euclidean generalized mean curvature with the additional bounded term \tr_{TS}\mathrm{II}, where \mathrm{II} is the second fundamental form of this fixed embedding. Indeed decompose a Euclidean test field into its ambient-manifold tangent and normal parts and differentiate that decomposition along TS. Its normal contribution is exactly the trace of \mathrm{II}. Consequently Euclidean monotonicity applies locally. Intrinsic and embedded density limits agree, since the embedding is isometric to first order. Monotonicity gives density existence and upper semicontinuity, including at nonreduced support points; their densities are at least one, by approximation by reduced points. Together with the clearing-ball upper bound from Lemma B.3, it gives uniform upper and positive lower area bounds on compact patches.

A common ambient model for varying metrics. We record carefully the uniform version when the metric changes. Suppose g_i\to g smoothly on a fixed patch, and all other data have the local bounds in the theorem. Any claimed failure of regular-point persistence or support convergence has a subsequence on which the failure persists. Pass further so that, on a slightly larger compact patch,

Set t_i=2^{-i} and take disjoint smooth bump functions \beta_i(t), equal to one on a neighborhood of t_i, supported in intervals of width comparable to t_i, and satisfying |\beta_i^{(b)}|\le C_b t_i^{-b}. After omitting finitely many terms,

is a smooth positive metric on a product neighborhood, with g_0=g and g_{t_i}=g_i. For each pair of fixed orders a,b, the summands after a spatial and b temporal derivatives are bounded by C_b2^{ib-i^2} for all sufficiently large i. These bounds, divided by any fixed power of t_i, still tend to zero. Thus all mixed derivatives extend at t=0. The plateaus also give \partial_tg_t=0 at t_i, so the slices carrying g_i are totally geodesic in G.

Extend this one smooth metric G across the edge of a smaller product patch to a closed manifold and isometrically embed it once. The images of the slices t=t_i converge smoothly to the slice t=0, and the second fundamental form of this single embedding is bounded on their common compact neighborhood. The preceding first-variation calculation therefore gives one Euclidean mean-curvature bound for all the embedded perimeter varifolds. Strict perimeter convergence from Lemma B.3 becomes varifold convergence in this same Euclidean space. This construction proves the needed sequential uniformity; it does not assume that independently chosen isometric embeddings have uniform bounds.

Exact minimizing tangent cones. Next consider a blowup of one phase at x\in S. In smooth coordinates, at scale \rho, the normalized data are

The metric here has been divided by the square of the physical scale: perimeter scales by \rho^d, volume by \rho^{d+1}, and this accounts for the factor \rho in the pressure. On every fixed window the data converge smoothly to g(x),\omega(x),0. The strict compactness lemma therefore identifies every subsequential tangent varifold with the unit-multiplicity perimeter varifold of a phase minimizing perimeter in the constant metric g(x). A constant weight can be divided out. After one linear isometry this is an ordinary Euclidean perimeter-minimizing phase. It is also a minimizing boundary current. Indeed, orient this Euclidean chart. A compactly supported integral current filling a boundary competitor has top dimension and is given by an integer-valued BV function q. The competitor is the boundary of \chi_A+q. Clipping that integer-valued function to [0,1] leaves it unchanged outside the compact discrepancy and gives a characteristic function without increasing variation. Any current competitor with smaller mass would therefore give a smaller-perimeter set competitor. This verifies the precise minimizing-boundary hypothesis of Simon [36, Chapter 7, Theorem 5.8]. All orientations in this argument are local.

The support lower bound makes this limit nonzero at the origin. Monotonicity based at x makes its mass ratio constant on all radii; the equality case, or the stationary radial identity in Simon [36, Chapter 4, Equation (3.6)], makes it conical. Equivalently, after a fixed local isometric embedding, dilations of the embedded ambient manifold converge smoothly to its tangent space, its second fundamental form is multiplied by \rho, and the same blowups give this cone. Its multiplicity is one as a perimeter varifold, although its density at the vertex need not be one.

Regularity and persistence under convergence. We now apply Allard’s graphical regularity theorem [2]. In its Euclidean form used here, fix an exponent q>d. For a rectifiable d-varifold of multiplicity at least one, on a ball centered at a support point and contained in the first-variation domain, sufficiently small excess of the normalized mass above one and sufficiently small

give a single C^{1,1-d/q} graph on a smaller ball [36, Chapter 5, Theorem 5.2]. A unit flat tangent supplies the small mass excess. The bounded mean curvature supplies the other small quantity, since it is O(\rho) under the local area bound. Thus a flat tangent makes the original point regular as a C^{1,\alpha} graph. The graph satisfies the weak prescribed-curvature equation H_g=p-\partial_\nu\log\omega; its coefficients and right-hand side are smooth functions of position, graph height, and graph gradient. The bounded slope gives uniform ellipticity. We spell out the passage from a weak graph to a smooth one. In a relatively compact chart write the equation as

Here A is the gradient, in the slope variables, of the smooth positive weighted graph-area integrand. Thus A_p is symmetric and uniformly positive on the bounded slope range; B is smooth in its arguments. For a compactly supported cutoff \eta, test translated differences with \eta^2\delta_hu. The difference of A has the form a_hD\delta_hu+c_h\delta_hu+b_h, where a_h is uniformly elliptic and all three coefficients are uniformly bounded. Since u is Lipschitz, \delta_hu is bounded. On the right move the difference quotient from B to the test function, using \|\delta_{-h}\phi\|_2\le\|D\phi\|_2. Ellipticity and Cauchy’s inequality then give

independently of small h. Hence u\in H^2_{\mathrm{loc}}. For v=\partial_ku, weak differentiation now gives

with coefficients evaluated at (x,u,Du). Both a and F are C^{0,\alpha} because u is C^{1,\alpha}.

To justify the divergence Schauder estimate for this weak H^1 solution, mollify a,F on a slightly larger ball and solve the coercive linear Dirichlet problems on the ball with the same trace as v. Their solutions v_\epsilon satisfy

where \lambda>0 is a common ellipticity constant. Thus they converge in H^1. Smooth-coefficient Sobolev regularity makes each approximant smooth in the interior; the local L^2–L^\infty estimate and then the a priori divergence Schauder estimate give uniform interior C^{1,\alpha} bounds [37, Lecture 6, Theorem 1 and Corollary 1; Lecture 18, Theorem 2; Lecture 12, Theorem 1]. These bounds use only the common ellipticity and Hölder bounds, not higher derivatives of the mollified coefficients. Passing to the limit gives v\in C^{1,\alpha}_{\mathrm{loc}}, hence u\in C^{2,\alpha}_{\mathrm{loc}}. Repeating the difference-quotient and weak Schauder argument for differentiated equations makes the graph smooth. The estimates are uniform for families with common C^{1,\alpha} graph bounds and uniformly smooth geometric data.

Likewise, near a regular point of a limiting phase choose a sufficiently small ball with mass ratio as close to one as needed, with strict slack in the Allard threshold, and with small scaled mean-curvature bound. Choose its radius also so that its sphere has zero limiting perimeter measure; almost every sufficiently small radius has this property. For smoothly convergent data, use the single ambient embedding construction above. Weak perimeter-measure convergence now passes the mass bound on this ball to the approximating phases. To center at a nearby approximating support point, decrease the radius by a fixed arbitrarily small factor: the recentered ball lies in the chosen ball for all large indices, and the strict slack pays the change in the normalizing power of the radius. Allard therefore gives one graph in each smaller ball for all sufficiently large indices. The graph equation and smooth data convergence then give smooth convergence on smaller compact patches. This also proves

for interior limits: a regular limiting point has just been excluded, and a point outside the limiting support is excluded by the uniform support lower bound and perimeter convergence. The same lower bound and weak convergence prove local support convergence; they also exclude stray support in compact subsets of phase interiors. The regular set is open in the support, so its complement \Sigma is relatively closed.

Transfer of the singular-dimension bound. For the singular-dimension transfer we use the content argument in Simon [36, Chapter 1, Theorem 3.26(2), and Appendix A, proof of Theorem A.4, step (1)]. Its hypotheses in the present application have all been established: on every compact coordinate patch and at every center, each vanishing scale sequence has a strict tangent subsequence; these tangents are homogeneous minimizing boundaries; singular sets commute exactly with translations and dilations; and, on compact windows of every such convergent sequence, singular sets eventually lie in every open neighborhood of the limiting singular set. For the last assertion, a contrary sequence of points has a convergent subsequence, contradicting regular-point persistence or support convergence. The data on the rescaled windows have uniform local derivative bounds, with derivatives of order j acquiring factors \rho^j and pressure derivatives acquiring \rho^{j+1}. Thus the compactness and regularity argument applies to all these sequences. We use only the cited content-transfer step, and spell it out next; no identification with the scalar-function family of Appendix A is needed.

Suppose m\ge8 and, for some s>m-8, the singular set in an interior window had positive \mathcal H^s measure. Let \mathcal H^s_\infty denote Hausdorff content, defined by covering with balls and summing their radii to the power s; normalization constants are immaterial. The Hausdorff-content density selection in the proof of Theorem A.4 provides an interior x\in\Sigma, scales \rho_i\downarrow0, and c>0 such that

Pass to a strict tangent limit C along these scales. The exact minimizing-boundary theorem gives \mathcal H^s(\Sigma_C)=0, because its hypersurface dimension is d=m-1 and its bound is d-7=m-8. Cover \Sigma_C\cap\overline B_{3/2} by finitely many open balls whose sum of sth powers of radii is less than c/2; the cover can first be chosen countable with this bound and then reduced to a finite subcover by compactness. Singular-set persistence puts every \rho_i^{-1}(\Sigma-x)\cap\overline B_1 in that same union for large i: otherwise a convergent sequence in its compact complement would give a singular limiting point outside the cover. Scaling of Hausdorff content now contradicts the preceding lower bound. This proves \mathcal H^s(\Sigma)=0 for every s>m-8. It is the density selection and this finite-cover passage, in addition to tangent information, that transfer the dimension bound.

If m\le7, the exact minimizing-boundary theorem says that every tangent cone is regular at its vertex and hence is a hyperplane. Flat-tangent regularity then makes \Sigma empty. Finally a nonflat minimizing cone is singular at its vertex, and translation invariance carries that vertex to each point of its spine. The spine is therefore contained in its singular set, giving the asserted spine bound. The preceding monotonicity argument, applied in the common embeddings for convergent data, also gives upper semicontinuity of densities at moving points by first passing to the limit at a fixed radius and then letting the radius decrease to zero. The smooth graph’s coorientation comes from the phase, and its first-variation equation is (B.9). All constructions are local in the base and require no ambient orientability or torus-dimensional regularity theorem.□

Proof. The metric, weights, and shifted pressure in the translated expanding windows are exactly the data pulled from B. Local compactness produces a subsequential limiting phase on B\times\R^d, and it is an absolute minimizer by Lemma B.3. The uniqueness and factorization of Proposition B.5 identify it with A_B\times\R^d. Every subsequence has this same possible limit, which proves convergence of the full sequence. Apply Theorem B.6 for the support and smooth assertions. The deeper-face statement follows by using the commuting product coordinates in Lemma A.3.□

Proof. Weighted perimeter is the area of the old torus product for invariant phases. The invariant normal variation with speed v is compactly supported in the smooth regular part and comes from an admissible base variation. Thus its energy has a minimum at parameter zero. For completeness the variation identities hold in every dimension: differentiating the induced metric gives \dot g_{ab}=2vB_{ab}, differentiating the unit normal gives \dot\nu=-\grad_Sv, and commuting the two ambient covariant derivatives in the derivative of B_{ab} gives, after tracing and differentiating the inverse metric,

The area derivative is Hv\,\dd\mu_S, while the pressure-volume variation is pv\,\dd\mu_S. The first energy derivative is therefore \int_S(H-p)v\,\dd\mu_S. At H=p the derivatives of the area measure and of the chosen speed multiply zero, and \dot p=v\partial_\nu p. Hence the nonnegative second derivative is

Weighted integration by parts gives

Compare the first variation in Chodosh and Li [9, Lemma 13] and the second-variation calculation in the proof of its Lemma 14. The displayed calculation proves the identity in the present stabilized setting; the stability and Gauss-equation mechanism is that of Schoen and Yau [35, Section 1]. The derivation here uses neither Chodosh and Li’s dimension restriction for smooth existence nor a regularity assertion for the torus product. Only the already smooth base patch and its invariant variations enter. The Gauss identity

gives (B.10). By (B.1),

which proves the final statement.□

Proof. For almost every t>0, delete \Omega\cap\{f>t\} from the minimizing phase. The removed boundary area is the area on S\cap\{f>t\}; the newly introduced boundary is the level surface of f in the interior of \Omega. The pressure-volume term increases by \int_{\Omega\cap\{f>t\}}p_s\,\dd\mu. Minimality gives the corresponding perimeter inequality. Integrating in t, the layer-cake formula and weighted coarea give (B.11). All comparisons are compact and occur on regular patches; the perimeter slicing formula justifies the same argument at exceptional critical levels by discarding a null set of levels.

Apply (B.11) to f=2(-p_s)_+v^2, initially by Lipschitz approximation. Its right side is bounded above by

Young’s inequality gives

The pressure estimate holds unchanged for p_s, and

Discard the resulting nonpositive multiple of p_s^2v^2. Thus

Since 2p_s\ge-2(-p_s)_+ on the boundary and H_{\rm out}=p_s, this is (B.12).□

Proof. The wall assertion follows by adding (C-2)|\grad v|^2 to Lemma B.8. For the bulk use Lemma B.9; the remaining gradient coefficient is C-3.05=.94. Apply the wall assertion to every cylinder slice and integrate in t. The horizontal coefficient is C-2=1.99, while the vertical coefficient is still C=3.99, so lowering the latter to 1.99 gives (B.16). The seam trace in the bulk estimate is precisely the matching trace in ().□

Proof. Compatible smoothing by Lemma A.4 gives the stated approximations: smooth the continuous clipped functions on compact parts and blend with their already prepared exact face copies. Allocate the error .1 beforehand over the finitely many requests on each face, and reproduce constants exactly on passive copies. For a packet of dimension m\ge8 and a set I of m-7 remaining labels, the smooth map h_I=(h_j)_{j\in I} is locally Lipschitz. Since \dim_{\rm H}\Sigma\le m-8, its image of \Sigma has zero (m-7)-dimensional Lebesgue measure. This follows on each relatively compact chart from the defining Hausdorff covers and then on the packet by a countable union. The set of threshold vectors for which t_I\in h_I(\Sigma) is null by Fubini. For measurability, intersect \Sigma with a compact exhaustion first: each image is compact, and their countable union is measurable. Give each interval (-.3,.3) its normalized Lebesgue measure. There are countably many packets and finitely many relevant subsets per packet, so choose a threshold vector outside the union of all exclusions. The same argument also works in a countable product of the threshold intervals: each exclusion is a null cylinder set depending on only m-7 coordinates. If fewer than m-7 labels remain, the asserted count holds without a condition. At a negative pole h_j\le-.9<t_j, and at a positive pole h_j\ge.9>t_j, proving the last assertions.□

Proof. Write \Theta(x) for the unweighted boundary density. First consider a tangent minimizing boundary cone C at a singular point. It is nonflat, since a flat tangent gives regularity. For any z\in C, the density ratio centered at z has limit \Theta_C(0) as its radius tends to infinity: balls centered at z are sandwiched between balls centered at zero whose radii differ by |z|. Consequently

Equality in monotonicity says that C is also a cone centered at z. Composing a dilation about z with the inverse dilation about zero shows translation invariance along z. Conversely, translation invariance along z gives equality of the two densities. The translation invariance directions form a vector subspace, the spine. Every spine point is singular: otherwise translation gives regularity at the vertex, and the cone is a plane. The minimizing-cone singular dimension estimate therefore bounds the spine dimension by m-8. These tangent-cone and regularity inputs are precisely those in Theorem B.6.

Fix a singular point x and a small number \eta'>0. There are d_x>0 and r_x>0 such that, at every radius r<r_x, the points

lie within \eta'r of some (m-8)-plane through x. If not, choose failing radii r_i\downarrow0 with density tolerances tending to zero. Rescale at x and pass to a tangent cone C. Fix a plane containing its spine. Failure for this particular plane gives points z_i in the rescaled unit ball a fixed positive distance from it. A convergent subsequence has limit z. Upper semicontinuity of densities under these convergent blowups gives \Theta_C(z)\ge\Theta(x)=\Theta_C(0). The preceding paragraph puts z in the spine, a contradiction. Smooth metric errors and bounded mean-curvature errors vanish in this rescaling.

Here is the countable decomposition, including its closure step. Group singular points according to lower bounds r_x\ge1/a and d_x\ge1/b, with positive integers a,b, and according to a density interval of length less than 1/(2b). Any two points x,z in the same group satisfy \Theta(z)>\Theta(x)-d_x. Thus the group itself has the plane containment at every one of its centers and every radius less than 1/a. Work within one buffered chart at a time. Intersect these groups with a countable family of compact subpatches Q\Subset O exhausting that chart’s portion of O, and take F=\overline{Q\cap\text{group}}. Relative closedness of \Sigma and the compact buffer ensure F\subset\Sigma\cap O. These compact sets still cover \Sigma\cap O.

To check the geometric condition at a closure center x, let x_i be group points tending to x and fix r<1/(3a). Apply the group estimate in B_{2r}(x_i). The planes pass through x_i, so a subsequence converges to a plane through x. Every group point in B_r(x) is in these larger balls eventually. Passing first to the limit of the planes and then to closure points gives containment with width 2\eta'r. Taking, for example, \eta'<\eta/3 proves (C.2); choose r_F below 1/(3a) and the buffered chart radius. This argument transfers the containment itself; no density continuity at a closure point is required.□

Proof. The polynomial q(y)=.29y^2-y+1 is strictly positive, and

Let y:(0,5\pi)\longrightarrow\R solve y'=q(y), with the two infinite endpoint limits, and put

At differentiability points of \delta, its gradient has norm at most one. Differentiating in \delta gives

which proves (C.4). Extending by the stated poles is continuous in the extended real line. Each finite clipping has a finite Lipschitz constant locally, including across either clipping interface.

Choose a continuous cutoff of \delta, equal to one on [\varepsilon,R_\varepsilon] and supported strictly in (\varepsilon/2,2R_\varepsilon). Multiplying it by 2.06\delta^{-2} gives a continuous nonnegative function supported away from F. Smooth it by nonnegative local convolutions and a partition of unity, with sufficiently small value error on this compact annulus and with support retained in a slightly larger annulus of the indicated size. The positive margins .06 on the central annulus and .06 below the upper bound give (C.5). The annuli lie in the strict anchor region for all small \varepsilon, by compactness of F.

At almost every point where the new maximum is finite, its gradient is the gradient of an active branch. The old branch obeys the old inequality. The new one obeys it because .58<a_r and D_\varepsilon\ge2\delta^{-2} wherever that branch is finite. The same reasoning applies at branch equality, using the elementary weak derivative rule for maxima of locally Lipschitz functions. The new core has closure inside the open positive pole region \{\delta<\varepsilon\}. By Lemma C.1 no old negative pole occurs in the support of the positive forcing. Positive and negative additions for the same label have disjoint strict anchor regions, so repetitions introduce no conflict. Indeed any opposite pole introduced later lies in the opposite strict anchor region and therefore cannot meet this support. Original poles and previously introduced opposite cores consequently remain protected throughout. Replacing the profile by its negative proves the other assertion.□

Proof. Put s=m-8. At scale r, the one-sided plane condition covers F\cap B_r(x) by at most C_m\lambda^{-s} balls of radius \lambda r, with centers in F, where \lambda=8\eta_0 can be made as small as desired. To see the bound, take a maximal \lambda r-separated family in the set. Projection to the approximating s-plane changes pairwise distances by at most 2\eta_0r; the projections are separated by a fixed multiple of \lambda r in an s-ball of radius 2r. Its elementary Euclidean packing bound gives the claim, with harmless changes in C_m and the covering radius. Absorb these fixed changes into the initial choice of \lambda. Choose \eta_0 so small that the resulting covering number is at most \lambda^{-(s+1/2)}.

Iterating the same covering at centers in F gives, for 0<t\le d below a fixed fraction of r_F, a covering by at most C(d/t)^{m-7.5} balls of radius t of the part of F controlling a ball of radius d. Here one first centers a ball of radius a fixed multiple of d at a nearby point of F and then iterates; the constants use only the fixed rough choice. Enlarging the final balls by a fixed factor proves the tubular-volume estimate

whenever \dist(x,F)\le d and the slightly enlarged ball is in the chart at allowed scales. Metric and Euclidean distances are uniformly comparable here. Dyadically splitting the regions 2^{-k-1}d<\delta\le2^{-k}d, and treating \delta>d separately, gives

Indeed the sum of the bounds is a geometric series with ratio 2^{-2}. In particular this density is locally integrable.

Choose a nonnegative smooth chart cutoff \chi, equal to one on a neighborhood of F, and, using Euclidean coordinates, define

For small \delta(x)>0, the part |x-y|<c\delta(x) has \delta(y) comparable to \delta(x); it supplies the lower bound for U. The same region supplies the upper bounds for U and its first derivatives. On the successive larger annuli use (C.8); their contributions have orders (2^k\delta)^{-1.5} and (2^k\delta)^{-2.5} respectively. The parts above the allowed local scale are bounded constants depending on F. Shrinking the final distance threshold absorbs them, and gives dimension-dependent constants c,C>0 such that

Set a=m-4 temporarily. For any Euclidean d'-plane T, direct differentiation of K(z)=|z|^{-a} gives

For d'=m or m-1 the last coefficient is positive. A sufficiently small, dimension-dependent metric distortion preserves a fixed positive fraction of it for traces in metric orthonormal frames. The connection and weight terms in the weighted Laplacian involve only first derivatives of U. On S_{\rm reg} the restriction formula adds its mean-curvature vector paired with \nabla U. This vector is locally bounded, since H_{\rm base}=p-\partial_\nu\log\omega by Theorem B.6. There is no dependence here on individual principal curvatures. Integrating the positive second-derivative kernel over |x-y|<c\delta(x) gives c\delta^{-3.5}. Equations (C.10), followed by another reduction of the distance threshold, therefore imply, in the ambient space and on the regular wall,

The leading constants again depend only on the fixed rough data. For example the first-order errors before division have size at most C_F\delta^{-2.5}, one power lower than the positive term. The differentiated kernels through order two are locally integrable off F: their strongest singularity is |x-y|^{-(m-2)}. Since the density is bounded near any such x and integrable elsewhere on its compact support, ordinary differentiation under the integral, with a vanishing small diagonal ball, proves that U is C^2 off F. No diagonal distributional term is present.

For a weighted domain Q, completing a square and integrating by parts gives the vector-field identity in inequality form

Choose a smooth cutoff \rho, equal to one near F and supported in the chart, and set A_b=-b\rho\nabla\log U in the bulk. On the wall use A_h=-b\rho\nabla_S\log U. For 0<b\le1, where \rho=1,

Outside a smaller neighborhood of F all errors from these fields are bounded by a piece-dependent constant.

On the cylinder take the horizontal field A_h and add the vertical component a(x)\partial_t, where

It has zero divergence in the vertical direction and satisfies 1.7a\ge .8|A_b| at the seam. The outward cylinder normal there is -\partial_t. Applying (C.12) with coefficients .8 and 1.7 on the two pieces, their boundary terms sum to

This explicitly permits arbitrary matching seam values. The vertical square costs at most C b^2\delta^{-2}+C_F in the cylinder interior. Choose b>0 sufficiently small, using the universal constants in (C.11), so that this is absorbed by half the positive term of order b\delta^{-2}. All remaining localization terms cost C_F\|v\|_2^2. This proves (C.6) on a joined side. The wall proof is the same calculation without the seam or the vertical field. Taking the smaller leading coefficient makes one c_0 valid for the wall and both sides.□

Proof. We give the parameter order first. Fix the rough number \eta_0 and coefficient c_0 from Lemma C.4. Below, constants in partition gradient bounds depend only on dimension and the fixed chart distortion. Choose \xi>0 so small that the rough estimate pays their total squared-gradient error with at most .04 of \mathcal N. Next choose a large fixed K, then a small fixed \alpha>0, and finally a strong flatness \eta\le\eta_0, as specified below. Only after the piece F has been selected for this \eta do we choose its small distance threshold and then \varepsilon_F. In particular C_F is never used to choose \xi, K, or \eta.

Put a_-=1/2, a_+=2e^{5\pi} and R_1=K\varepsilon. The shell supporting D_\varepsilon lies in a_-\varepsilon<\delta<a_+\varepsilon. Choose a smooth angle \vartheta(\delta) equal to zero for \delta\le a_+\varepsilon and equal to \pi/2 for \delta\ge\xi R_1, with

Such a function is obtained by a smooth transition in \log(\delta/(a_+\varepsilon)) of length at least a fixed multiple of 1/\xi. Thus it suffices to choose K\ge (a_+/\xi)\exp(C/\xi), independently of the piece. Set q=\cos\vartheta and q_\infty=\sin\vartheta. They extend as constants across F, and |\nabla q|^2+|\nabla q_\infty|^2\le\xi^2\delta^{-2}.

Take a smooth normalized squared partition \sum_i\phi_i^2=1 on chart boxes of scale R_1, with uniformly bounded overlap and \sum_i|\nabla\phi_i|^2\le C_mR_1^{-2}. One construction translates a fixed compactly supported bump on a Euclidean lattice and divides each bump by the square root of the sum of their squares. Restrict the same ambient functions to the wall and freeze them on the cylinder. Only finitely many boxes meet \supp q, which is compact in the chart at the scales in use. The family q\phi_i,q_\infty is a squared partition, and its total gradient square is at most C_m\xi^2\delta^{-2} on \{\delta<\xi R_1\} and zero elsewhere. The estimate follows from \delta<\xi R_1 wherever q\ne0.

For a squared partition \sum_i\chi_i^2=1, direct expansion gives

where a(x) is .8 on bulk and 1.7 on wall or cylinder. The mixed terms vanish because \sum_i\chi_i\nabla\chi_i=0. Choose \varepsilon_F small enough that \xi R_1<d_F. Equation (C.6) and the choice of \xi now give

For example it is enough to impose 1.7C_m\xi^2/c_0\le.04; the other partition term is nonnegative and has been discarded. Lipschitz distance cutoffs are allowed by approximation on the regular test supports.

Consider one box meeting the support of q. Choose a nearby point of F and apply (C.2) at a radius L_mR_1, where a fixed L_m includes the box and every nearest point of F needed in it. Taking \varepsilon_F smaller ensures this is below r_F. Let L be the resulting (m-8)-plane and define

The Euclidean Hessian of P is an orthogonal projection of rank eight. Its trace on every (m-1)-dimensional subspace is at least seven. Its gradient has size at most \zeta, and throughout the box \zeta\le C_mR_1 (after fixing \alpha and K).

For either the ambient or the tangential weighted derivatives, the vector field A=d_1\nabla\log\zeta has potential

Let \tau bound the chart’s metric distortion. The rank traces have error at most C_m\tau, and |\nabla P|^2/\zeta^2\le1+C_m\tau. The connection, weight, and wall mean-curvature contributions to \Delta_\omega P have size at most C_FR_1. Thus one may arrange C_m\tau\le .005 and C_FR_1\le .025, uniformly in the box and its plane. Equation (C.16) then gives

Indeed the respective Euclidean lower bounds are 8-3=5 and 2\cdot7-8=6. Only the mean curvature of the wall enters this calculation, so these bounds remain valid arbitrarily near its singular set on the regular test supports. The positive term \alpha^2\varepsilon^2 also makes the fields nonsingular along L; the estimates hold inside the inner shell as well.

In the bulk use A_b=\nabla\log\zeta. On a joined cylinder use A_c=2\nabla_S\log\zeta+\beta\zeta^{-1}\partial_t, where \beta=(.8/1.7)\sqrt{1+C_m\tau} is chosen so that \beta/\zeta\ge(.8/1.7)|A_b| at the seam. The metric distortion determines this fixed upper factor. The vertical component has zero divergence and is orthogonal to the horizontal component. Its only interior cost is \beta^2\zeta^{-2}. The boundary terms in (C.12) are nonnegative after summing, by exactly the cancellation used in Lemma C.4. After multiplication by the coefficients in (C.1), the limiting interior coefficients are

and 1.7\cdot6=10.2 on the wall alone. With the explicit error bounds just imposed, the coefficients are at least .8\cdot4.9=3.92 in bulk, 1.7\cdot5.9-(.8^2/1.7)1.005>9.6 in the cylinder, and 1.7\cdot5.9>10 on the wall. In particular all exceed 3.8. Applying the vector inequality to q\phi_iv gives

This is a bound throughout the localized support; its nonnegative right side may therefore be restricted to the shell.

We finally compare \zeta with \delta on that shell. A nearest point z\in F lies in the ball used for the plane condition, so

The first factor is ensured by the fixed chart accuracy and its buffer, after shrinking the scale so that a minimizing short path to F stays in the chart. Since \delta\ge a_-\varepsilon there, choose \alpha/a_-\le .005 and then L_m\eta K/a_-\le .015. These choices give \zeta\le1.03\delta. Notice that K was already fixed using the rough coefficient, and that the strong flatness was chosen before selecting F. Since 3.8/(1.03)^2>3.5, (C.17) controls 3.5\delta^{-2} on the shell. Summing over boxes uses q=1 there and \sum_i\phi_i^2=1, and then (C.15) gives

Together with (C.5), this proves the stronger coefficient bound

This yields (C.13). A compact test on a pure cylinder can be translated to positive height in a joined cylinder and extended by zero onto the bulk, so it obeys the same estimate.□

Proof. We give the estimates for tests first; they extend to the completions by density. On the enlarged window the weights are uniformly comparable to a positive constant, the ambient geometry is uniformly smooth, and the unweighted base mean curvature of the regular walls is bounded. The latter bound follows from the prescribed weighted mean curvature and the bounded first derivative of the logarithm of the weight. The perimeter varifolds have density at least one and no additional boundary term on this interior window, by Theorem B.6. They therefore meet the hypotheses of the Sobolev inequality used below. No bound on the individual principal curvatures is needed.

We use the submanifold Sobolev inequality in the following precise form. For a d-dimensional Euclidean rectifiable varifold of density at least one with generalized mean curvature H, and a nonnegative compactly supported test f,

This holds in arbitrary codimension; see Simon [36, Chapter 4, Section 5, Theorem 5.7]. This is the submanifold Sobolev inequality of Michael and Simon [31]. A smooth extension of the fixed ambient chart metric to a compact manifold and a fixed smooth isometric embedding [33] give the corresponding local inequality in our charts. The extra mean curvature of the embedding is bounded on this fixed compact geometry. For the moving-domain assertion the embedding is chosen once in the fixed ambient geometry; its constants are therefore independent of the wall in the sequence. Applying Equation (D.3) to a power of the test gives the usual H^1 Sobolev exponent on the wall. Applied to S_{\rm reg}\times\R, it gives, with

a local L^q bound on the cylinder. A test on the half-cylinder is reflected evenly at its base before this application. The wall alone has the stronger exponent 2(m-1)/(m-3)>2.

Here and below cutoffs are taken on the enlarged window and are pulled back to both seam sides. The cutoff errors are bounded by the local L^2 norm. Normalize \mathfrak h_Z(v)\le1. Choose a cylinder cutoff which is one at s=0 and vanishes at the outer end of a fixed bounded interval. The fundamental theorem of calculus and Hölder give

The last step uses the cylinder L^q bound and 2(b-1)=q.

The bulk need not have a uniform smooth boundary, so we use a BV calculation there. Extend \abs{v_{\rm bulk}}^b by zero in the ambient chart, including its compact cutoff. For an initial test its support misses the singular set, and the BV jump formula gives precisely the boundary contribution \int_{S_{\rm reg}}\abs{v(0)}^b in addition to the interior variation. This is also obtained directly by integration by parts against unit vector fields on the regular side. The ambient L^1 Sobolev inequality and Cauchy–Schwarz therefore yield, for J=\int_\Omega\abs v^q,

Here bm/(m-1)=q; the first integral on the right is at most J^{1/2}\norm v_2, because b=(q+2)/2. Since (m-1)/m>1/2, Equation (D.5) bounds J. Rescaling removes the normalization. Thus all pieces have a local L^p bound for some p>2 controlled by their matching form norm. This argument extends a power as a BV function; it constructs no H^1 extension operator for the bulk.

We next exclude concentration in the moving version. On a slightly larger compact window let F be the compact portion of the limiting singular set. Its ambient volume and its limiting perimeter measure are zero. For a closed \delta-neighborhood of F, perimeter measure convergence and the closed-set upper bound for weakly convergent measures imply

with a fixed outer compact cutoff if necessary. The right side tends to zero as \delta\downarrow0. The same conclusion for bulk measure follows from ambient volume bounds, and for a cylinder follows by multiplying the wall measure by the bounded interval length. Hölder and the local L^p estimates give

Thus the order is first i\to\infty at a fixed singular neighborhood, then \delta\downarrow0. Neither disappearing components nor a thin boundary layer can retain L^2 mass.

Off F, graphical convergence and the absence of stray wall support supply smooth charts for both bulk phases and their seams. Rellich compactness in those charts gives the asserted weak H^1 and strong L^2 limits. The trace map on each fixed regular chart is bounded and linear; weak convergence of both traces preserves their equality. Combining these regular limits with the no-concentration conclusion proves all the stated L^2 and bounded-potential convergence. Exhausting the regular pieces gives lower semicontinuity of their gradient integrals.

It remains to justify membership in the limit completion, rather than assume it. On this one limiting geometry, monotonicity and the local area bound give

for sufficiently small balls in the fixed window. Since \dim_{\rm H}\Sigma\le m-8<m-3, a compact singular portion has ball covers, inside arbitrarily small neighborhoods, with \sum_i a_i^{m-3} arbitrarily small. For cutoffs equal to one on these balls, supported on their doubles, and with gradients at most C/a_i, the maximum of the cutoffs has squared-gradient integral at most

on the wall and on bounded cylinder intervals. The bulk cost is at most C\sum_i a_i^{m-2} and is smaller. A finite subcover suffices. Use the same base cutoff on the bulk and cylinder, preserving seam matching. First truncate the limit function to a bounded function; then multiply it by one minus these cutoffs. The derivative of the cutoff has vanishing energy after truncation, and the other error tends to zero by absolute continuity of the already finite gradient integrals. Its L^2 error tends to zero as well. The resulting functions miss the singular set. Matching traces glue their two H^1 pieces across every regular seam chart; ordinary mollification in such charts, and a partition of unity, approximate them by the defining tests. This proves membership and Equation (D.2). Capacity is used only on the limit; no cheap cutoff on every approximating singular set, and no convergence theorem for the entire moving Sobolev spaces, is needed. The same proof applies to pure cylinders and product limits. For a product with a extra coordinates, use ambient dimension m+a and singular-dimension bound m+a-8 in the preceding exponents; the strict codimension needed for the capacity estimate is unchanged. Thus this argument supplies exactly regular graphical identifications, strong L^2 convergence without singular concentration, matching seam traces, and the lower semicontinuity used in the later form transfers. It does not assert smooth convergence at singular points.□

Proof. First \int A_\eps v^2\to0 for every fixed v\in\mathcal H_Z. Indeed Equation (D.7) bounds these forms uniformly by a constant times \mathfrak h_Z, whereas for a defining test the integral is zero for small \eps, since its support misses F. Density and Cauchy–Schwarz for the nonnegative form give the assertion.

We detail the localization needed on noncompact pieces. If \sum_\alpha\chi_\alpha^2=1 is a locally finite Lipschitz partition which has the same seam traces, then

The mass term e_0\norm v_2^2 allows the last integral to be at most \tau\mathfrak h_Z(v) for any prescribed \tau>0. For example use a sine/cosine pair of a Lipschitz cutoff angle, equal to \pi/2 on a fixed compact base set containing \supp A_\eps for all small \eps, and zero outside a sufficiently large compact neighborhood. Its transition width makes its derivative arbitrarily small. The cosine part sees only V. Its bound is the assumed global form inequality for V; no compact support assumption on V is needed. The sine part has compact base support, where the old potential is bounded. On a joined side also use a squared partition into cylinder intervals of a sufficiently large fixed length. The first member is identically one near s=0 and extends as one on the bulk; all the others vanish near the seam and extend as zero there. Their squared gradients have an arbitrarily small sum. All cutoffs in the base are pulled back to the cylinder.

Choose the partition error so small that a violation of Equation (D.8) forces one localized test to have quotient larger than \max\{\theta,.78\}+\eta/2. This follows directly by summing the potential integrals and Equation (D.9); for instance it suffices to have (\max\{\theta,.78\}+\eta/2)(1+\tau) <\max\{\theta,.78\}+\eta. The selected test has compact base support and a bounded cylinder interval. Passing to a subsequence, its interval either stays in a fixed bounded range, or escapes to infinity. In the latter case it is translated to a fixed interval on the pure cylinder. Both the old bound and Equation (D.7) hold there by translating its tests into the original joined side. Thus in all cases the selected sequence lies in one fixed Hilbert space, in a fixed compact window.

Suppose now that \eps_i\downarrow0 gives such violating tests v_i, normalized by \mathfrak h(v_i)=1. By Lemma D.2 and weak Hilbert compactness, v_i\rightharpoonup v in this fixed space and v_i\to v strongly in L^2. Write w_i=v_i-v. Then

The bounded old potential satisfies \int Vv_i^2\to\int Vv^2\le\theta\mathfrak h(v). The new form vanishes on v, and its mixed term vanishes by Cauchy–Schwarz and the uniform bound on w_i. Consequently Equation (D.7) gives

This contradicts the selected quotient. There are only three raw pieces, so the same sufficiently small scale works for all of them.□

Proof. Only finitely many occurrences are involved by Definition 5.1. Here and below a source is extended along the directed face-to-coface relations, through their finite depth. Closing under arbitrary alternating face and coface walks is neither required nor performed. For a common normal coordinate take the largest of the finitely many requested starts, so commuting chains give identical occurrences. It is enough to prove the assertion for any sequence of their fronts with \min_iR_i\to\infty, passing to a fixed occurrence and piece if needed. Choose every localization width below using \eta and e_0 first, independently of the front positions.

Separate the new support from its complement with a sine/cosine pair as in Equation (D.9). More explicitly, an angle can be \pi/2 times a product of a compact base cutoff, equal to one on K, and normal cutoffs which equal one on [R_i,\infty) and vanish before R_i-L. Enlarge the compact base window and L until the sum of squared derivatives is sufficiently small. Extend the angle by zero outside the product sector. Taking the fronts beyond all collar thresholds makes this extension well defined. The cosine test misses the addition and has the old bound \theta. Partition the sine part further into fixed large boxes in every normal coordinate, and into fixed large intervals in cylinder height when applicable. The total localization error is as small a fraction of \mathfrak h as required.

If Equation (D.11) failed along fronts tending out, the same quotient selection as in Lemma D.3 would give normalized tests on selected boxes with quotient at least \beta+\eta/2. Such a box must meet the new potential support, since otherwise the old bound \theta\le\beta applies. Its center therefore has

Its base window is fixed and compact. Recenter every normal coordinate. A bounded cylinder interval stays in a bounded-height joined window, or, after another subsequence, is translated to a pure cylinder. The ambient geometry on the recentered windows is exactly the product geometry, and the true cuts converge to those over B times \R^d. The old potential is exactly V_B\circ\pi_B there, because the fronts also exceed its fixed product thresholds. The new potential is bounded above by the fixed bounded function D\circ\pi_B.

Apply Lemma D.2. For its limit v,

The face form bound applies to the right side by slicing the \R^d coordinates. For defining tests this is Fubini’s theorem: the gradient terms in the extra directions are nonnegative, and all coefficients .8,1.7,e_0 are the same in every base dimension. Approximation in the limit completion, proved in Lemma D.2, gives the result for v. In a pure-cylinder limit use the face joined bound after translation in cylinder height. Thus the right side of Equation (D.12) is at most \beta, a contradiction. Finite incidence permits simultaneous choices for all occurrences. This limit compares bounded potential integrals on moving domains; it uses no subtraction of vectors belonging to different form spaces.□

Proof. Consider a plus addition on B, with forcing profile u and shell potential D. Choose a compact base window K\Subset\{h_j>t_j\} containing both \overline{\{u> -\infty\}} and \supp D in its interior. The first set includes the inner pole region; keeping only \supp D, which is annular, would not suffice for this extension. In a coface product over K put

Each cap w_i is -\infty up to a late front, then crosses to +\infty, and remains there, with

on its finite band. For example integrate 2w_i'=.58w_i^2+p_j^* on the finite band and extend by its poles. The crossing length is 2\pi/\sqrt{.58p_j^*}<\infty; finite truncations remain locally Lipschitz at both ends. Only this label’s positive floor is used. Each smooth cutoff \chi_i takes values in [0,1] and finishes increasing to one before w_i begins its crossing. Choose these cutoff fronts to satisfy Lemma D.4 for the already fixed D and shell scale. Outside this product occurrence use the inactive value -\infty. The old profile is replaced by its maximum with u_E.

For locally Lipschitz finite truncations, a finite minimum or maximum has the derivative of one active branch almost everywhere (on a tie, the branch derivatives agree almost everywhere on the tie set). If the active finite branch is u, every cap has already left its -\infty plateau; hence every \chi_i=1 and the entire shell potential D is available. If it is a cap, its derivative cost is paid by the preexisting baseline:

The old branch retains its old inequality. Thus no sum of normal cap derivatives, and no extra cap potential, enters the budget. For a minus update negate u_E and take a minimum with the old profile. The strict anchor regions for opposite signs are disjoint and contain no opposite old poles by Lemma C.1.

Write T_i^+ for the end of the ith cap crossing. Enlarge the plus core by the open set

The properly embedded closed tails ensure that its closure in the coface has \delta_B\le\varepsilon/2 and t_i\ge T_i^++b_i. It is therefore contained in the open set \{\delta_B<\varepsilon\}\times\prod_i(T_i^+,\infty), where every branch of the minimum is +\infty. This proves closure confinement, including at the cap fronts, with the corresponding statement for a minus core. The front positions are also chosen beyond the exact anchor-product thresholds, so its support retains the designated strict anchor sign.

Use the same caps and cutoffs for the same normal coordinates in every identified occurrence. On a face obtained by hitting one of these coordinates, that coordinate’s cap is eventually +\infty and its cutoff is one, and that factor disappears from Equation (D.13). If an unrelated coordinate is hit, an intersection sufficiently far in its collar would require x to escape into a further face of B, which the compact support forbids. Finite incidence allows all relevant intersection thresholds to be exceeded at once. These observations prove compatibility for all chains and show why a commuting product occurrence is not counted more than once.□

Proof. First consider the packet system after the radial cut. Process its finite set of depths from the deepest faces upward. The anchors and thresholds of Lemma C.1 are fixed once, before any updates. Initially the cumulative addition is zero. Maintain a single common upper form bound at least .78; it is harmless to start this bookkeeping bound at .78 although the initial form vanishes. Enumerate the countable pairs consisting of a packet and a possible successive local addition number by distinct integers \nu\ge1. Reserve \gamma_\nu=.1\,2^{-\nu} for that pair, allocating at most half to Lemma D.3 and half to the simultaneous extensions of Lemma D.4. The common bound after any collection of completed updates is at most

We may consequently take \theta_*=.88. There is no requirement that a shell scale or its extension fronts be uniform in the packet or in the addition number.

Suppose all strict further faces of E have been treated and their updates extended. The remaining risky set

is compact. To prove this, if points in it escaped, the packet escape property Lemma A.3 would give a subsequence over a deeper face B, with compact convergent base coordinates and every escaping normal coordinate tending to infinity. The updated cores are exact products there. There are finitely many relevant labels, so pass to a further subsequence with a fixed subset of m-2 undecided labels. Membership in each of their closed sets G'_j passes to the base limit. Thus the limit base point has at least m-2 undecided labels. This implies membership in H_B^*, because \dim B-2\le m-2. By Corollary B.7 the translated true cuts converge to the true cut over B times the normal Euclidean factor. Support convergence puts the limiting base point on the true boundary over B. A regular limiting boundary point would have a neighborhood of smooth graphical convergence, which contains no singular points of the approximating walls. The limit must therefore be singular on the face. This contradicts its completed shielding. Thus K_E has no escaping sequence; it is closed, and the compact-part property proves compactness. If m<8 it is already empty.

Denote this initial compact set by K_E^{(0)}. For each remaining label and each strict anchor sign, cover K_E^{(0)} in that open region by the countable compact flat pieces of Lemma C.2. These pieces need not themselves be an open cover. Enumerate the signed pieces; their eventual inner forcing neighborhoods will provide the open cover needed for termination. For each, choose its shell sufficiently small by Lemma D.3, then hold that scale fixed and extend its caps, potential, and core by Lemmas D.4 and D.5. Supports can be as small as needed to stay within the prescribed strict anchor region. Opposite updates for one label therefore never conflict. Profiles retain Equation (D.14) by their active-branch inequalities, and cores enlarge monotonically.

To prove finite termination, after N additions set

Each O_N is open: with finitely many labels its complement is a finite union of intersections of closed G_j^{(N)}. Core enlargement implies O_N\subset O_{N+1}. Fix x\in K_E^{(0)}. For each of its strict anchors, some enumerated compact piece contains x, and the addition on that piece puts x in an open enlarged core, irrespective of how small its positive shell scale was chosen. Only finitely many anchors are involved. Once those additions have occurred, only threshold-equality labels can remain undecided at x. By Lemma C.1 there are at most m-8<m-2 of those. Thus x\in O_N for some finite N. The increasing open sets O_N cover K_E^{(0)}, so compactness supplies a finite subcover; taking its largest index gives K_E^{(N)}=\varnothing. Stop there. This argument proves an actual finite stopping time before forming any infinite sum of profiles. All subsequent updates only shrink the guard.

There may be countably many packets at one depth. An originating packet has finitely many directed coface occurrences, so each individual extension has already been chosen in finitely many places. Conversely, a fixed target packet has finitely many source faces, including through chains, so only finitely many packets at that depth affect it. Each of those sources required finitely many additions. This uses both directions of finite incidence from Definition 5.1. Its updated functions are finite sums or finite minima and maxima; there is no infinite limiting profile on that packet. Its product thresholds are bounded by finitely many chosen thresholds, and all its potential derivatives remain bounded. The common form bound after a whole depth follows from the same summable allowance. This justifies proceeding to the next depth, and proves the asserted smoothness, compatibility, and form bounds throughout the system.

The new guard is closed because the number of labels is finite and each G'_j is closed. On a far collar, exact product cores give the same count upstairs and downstairs. Lower face dimension weakens the required count, proving the projection assertion in (ii). These are precisely the closed-set and product-limit properties needed for the guarded regularization in Appendix E; a quantitative smooth tube is obtained there from regular graphical convergence.

For the initial radial cut, process its locally finite compact transition shells in order, choosing the supports of all additions inside buffered neighborhoods of their respective shells. By Lemmas 5.4 and B.4, the barrier subbands separate these neighborhoods, and the true boundary lies in their interiors. The true singular set in a shell is closed inside its compact buffered annulus, and the guard is closed. Their intersection is therefore compact. Apply the same increasing-open-cover argument there, keeping the closure of every forcing region inside this one shell buffer. There is no coface extension at this step. Before any addition only finitely many previous additions have been made, so their total is a bounded compactly supported potential and Lemma D.3 applies. The base localization is obtained from a proper distance cutoff; all estimates after localization use only fixed compact data. Use summable allowances as before. Local finiteness leaves smooth potentials and well-defined profiles, and a compact test sees only finitely many shell additions. The form bound therefore persists after all shells are treated. This proves the radial assertion and completes the construction.□

Proof. On a compact set, closedness of H_E^* and its disjointness from the singular set give a regular neighborhood of every point where the guard meets the boundary. A finite subcover and the normal inverse function theorem give a positive tube radius and all the asserted finite-order bounds there. The same compactness argument applies to the separation of the current core closures, which lie inside the barrier domains of Lemma B.1 and Theorem B.6 and whose neighborhoods miss the support.

We explain why these statements remain uniform at infinity. If a sequence escapes in E, the finite face description in Lemma A.3 gives, after passage to a subsequence, a product window over a further face F in which the F coordinates converge in a compact set and all the remaining normal coordinates tend to infinity. Bounded metric balls lie in these product windows eventually. The true phases converge there to the product of the true phase of F, by Corollary B.7. The updated cores and their complements are exact products. Consequently a limit of guarded points lies in H_F^*: a count of at least m-2 is at least \dim F-2. The limiting boundary point is therefore regular. Smooth, single-sheet convergence near this point supplies an embedded normal tube of a fixed positive radius, and excludes additional support in a smaller neighborhood. In particular no sequence at distance tending to zero from both the guard and the support can have degenerating tube radius or unbounded prescribed finite derivatives. This proves the first assertion by contradiction, with buffers reduced if necessary. The barrier statement at infinity follows in the same way from its face statement and phase/support convergence.

For uniform convergence in an entire elementary collar, suppose a discrepancy persists at (x_i,t_i) with t_i\to\infty. If x_i stays in a compact part of B, local product convergence gives a contradiction. Otherwise descend through the finite face diagram of B. Commutation of its collars with the t collar puts both the partition of E and that of B over the same further face F, with compact tangential coordinates and all escaping coordinates tending to infinity. Both limiting partitions are the same unique product partition. Local support convergence implies convergence of truncated distances: a nearest point at distance less than 1 is detected in a slightly larger fixed ball, and if there is no such point both truncated distances approach the saturated value. Away from the limiting support, phase convergence fixes the sign. This proves uniform C^0 convergence. On a guarded tube the limiting boundary is regular; smooth graph convergence and the normal inverse function theorem give C^J convergence of signed distance on smaller tubes.

Choose radii from the deepest faces upward. Each coface radius is smaller than a fixed fraction of every relevant further-face radius. For example, a point within \rho_E of H_E^* projects, far out, to within 2\rho_E of H_B^*, and |\varsigma_E|<\rho_E implies |\varsigma_B|<2\rho_E there. These are statements about the fixed true geometry and exact cores, without approximation tolerances. This order also handles a simultaneous escape in several coordinates and proves the last assertion. At the radial step the same proof on compact blocks requires no assertion about an initial spatial end.□

Proof. First choose all the geometric radii in Lemma E.1. Prescribe positive value and C^J error targets on the guarded regions, small enough also to preserve the core signs. Allocate smaller targets to faces to meet the requests of all their cofaces, including through chains, as in Lemma A.4. There are only finitely many such requests per face and finite depth. This allocation does not change any tube radius.

Work now from deeper faces upward. Smooth \varsigma_E by local convolution and a smooth partition of unity, with arbitrarily small value error everywhere and derivative error on the regular buffers. These errors can be prescribed separately: use the smooth function itself on an inner regular buffer and blend to a value approximation outside a larger one. On a far collar blend to the previously chosen face function. Lemma E.1 makes the discrepancy from the face’s true signed distance arbitrarily small by moving the collar start. The discrepancy of the already chosen face function from that distance is instead paid from its preallocated error target. These two errors are kept separate.

More explicitly, take cutoffs f_i from 0 to 1 in the elementary normal coordinates, in valid far product ranges. Use weights

They sum to 1. All derivatives through order J are bounded by constants fixed before the targets are allocated (translated cutoffs have the same bounds). Thus blending costs a fixed multiple of the participating errors. Face copies agree on all relevant overlaps once the starts exceed their product thresholds. When any f_i=1, every participating copy equals that face copy on the overlap; hence the result is exactly compatible. This proves the claimed approximations.

Near a possible zero in a guarded tube, the defining function has nonzero normal derivative and its zero is a small normal graph over the true wall. Off the tube, uniform value closeness preserves the phase where its sign is required. The copied end functions already have regular zeros. On the remaining compact region, a relative regular-value perturbation makes all zeros regular without changing the copied ends. For completeness, if g is the preliminary function, choose a smooth \chi\ge0, positive on its nonregular zeros and zero near the regions to be kept fixed. For an arbitrarily small regular value c of -g/\chi on \{\chi>0\}, the function g+c\chi has regular zeros there. Where \chi=0, its differential is that of g, since d\chi=0 there. Bounded derivatives of \chi and small c preserve all strict error and sign requirements. On the initial space use a positive smooth \chi with locally prescribed small derivatives and the same regular-value argument; the radial separators are preserved.

If the perturbed wall is t=\alpha(x) in a true normal tube, the map

with \chi_1=1 on the needed graph region and supported inside a larger regular buffer is a diffeomorphism for sufficiently small graph norm. It carries the graph to t=0, preserves its two local sides, and tends to the identity in C^{J-1}. Choose the buffer cutoff on the regular wall slightly beyond the projection of the smaller guard neighborhood and extend by the identity. Uniform regular charts give bounded cutoff derivatives. The same formula restricts to the wall used as a cylinder base. No product property of this auxiliary comparison is asserted or needed.□

Proof. For a side packet \dim A=m, a point in D_A projects to at least m-1 remaining undecided labels, hence into H_E^* and the parent’s old required set. For a hit packet \dim A=m-1, it projects to at least m-2 remaining undecided labels. The wall lies in G_{j_0}, which supplies the additional old undecided label. The projection is again in both required sets. These inclusions are unchanged on the fresh cylinder, where all relevant data are pulled back from its base. All these maps decrease lengths in the raw metrics; therefore a small neighborhood of D_A projects into small neighborhoods of the two parent sets. The closed-set count, rather than a continuity claim for the counting function, is what is used here.

On compared true pieces in a sufficiently small neighborhood of the old required set, continuity and the bounded derivatives of the fixed parent data give

By Proposition B.10 and Definition B.11, the raw form on these pieces dominates R-1.02\Pi_{j_0}, with spare gradient coefficients .94 on bulk and 1.99 on a wall or cylinder. The bound in Proposition D.6 pays for the additional potential 1.16\sum_{j\ne j_0}D_j from these spare gradient terms and a mass loss of at most .9e_0. Since b_r>1.02, dropping the remaining nonnegative current-label term leaves

Indeed the unrounded gradient remainders are .94-.9\cdot.8=.22 and 1.99-.9\cdot1.7=.46, and the mass loss is at most 1.9e_0. The exact numerical margins are

These identities use 0\le t_r<t_r+\Delta_r\le1. Thus Equation (E.2) exceeds Equation (E.1) by at least 98e_0 in mass and .2 in gradient, in addition to strictly positive potential margins.

Use the comparison diffeomorphism of Lemma E.2 on bulk and on the common wall/cylinder base, leaving the cylinder coordinate and torus variables unchanged. A test supported in a smaller buffered neighborhood of D_A becomes a compact matching test on the true regular pieces, extended by zero outside the comparison region. For a test in one component set it to zero on other components. In the bulk away from the wall the map is the identity and the phase signs already agree. All supports stay away from singular points and their cylinders.

The pulled-back metric, volume weight, potentials, intrinsic stabilized wall curvature, and outward mean curvature converge uniformly with the chosen finite derivatives. For example the induced scalar curvature can be computed from Gauss’ equation, so the second fundamental form and the ambient curvature give the requisite bounds. Consequently, for arbitrarily small \delta>0 the discrepancy of the compared forms is bounded by

The last term is controlled using the first unit interval of the raw product cylinder:

This follows by writing u(0)=u(t)-\int_0^t\partial_tu and integrating in t. It needs no zero boundary condition. Choosing all approximation errors sufficiently small leaves, for example, a reserve of 90e_0 in mass and .15 in gradient. These choices can be prescribed on parent buffers before the individual new components are known: the projection inclusions place every relevant point there. At the radial step the enlarged compact blocks supply the bounds; preserved separators trap each new component in one such block before cylinder attachment. Finally impose all coface requests through the finite tolerance allocation already used in Lemma E.2.□

Proof. In a normal collar of a smooth wall write the old stabilized metric as ds^2+\beta_s for s\le0, where s increases outward. Here \beta_s includes the old torus factors. The raw joined metric is ds^2+\beta_{\min(s,0)}. Choose a smooth concave function \Phi equal to t for t\le-1 and to 0 for t\ge1, with 0\le\Phi'\le1. Such a function is obtained by integrating a smooth decreasing step function symmetric about 0. It satisfies \Phi\le0 and \int\Phi''=-1= -\int|\Phi''|.

For a smooth positive function \epsilon(x) on the wall put

It equals the raw metric outside |s|<\epsilon(x) and differs from it in C^0 by O(\sup\epsilon). In a fixed tangential chart the extra derivatives of \phi satisfy

Choose \epsilon=\zeta^2 with \zeta>0 as small in C^2 as prescribed. Then |d\epsilon|^2/\epsilon=4|d\zeta|^2 and d^2\epsilon=2\zeta d^2\zeta+2d\zeta\otimes d\zeta. Thus these tangential errors tend to zero. On a packet, positive compatible choices have a positive infimum and bounded derivatives by the compact-part/product-end description. Their size, but not the resulting higher metric bounds, can be made arbitrarily small. Normal collar widths are also positive packetwise by the same description.

Let B^{\rm old}_\tau,H^{\rm old}_\tau be the slice second fundamental form and mean curvature of ds^2+\beta_s. Exactly,

For a metric ds^2+\beta(x,s) the Gauss and normal variation identities give

The tangential derivative bounds just proved show that all terms on the strip are uniformly bounded except

In particular the formula differentiates H_s only in s; potentially large mixed xs derivatives do not give another scalar spike. The remaining contribution from this derivative is -2\phi_s^2\partial_\tau H^{\rm old}_\phi, which is bounded.

Let J_\epsilon(x,s) be the density of \beta_\phi relative to the zero slice. Uniformly on the strip, J_\epsilon=1+O(\epsilon) and H^{\rm old}_\phi=H^{\rm old}_0+O(\epsilon). For each fixed x,

Writing E_x=\int_{-\epsilon(x)}^{\epsilon(x)}|\partial_su|^2ds, the fundamental theorem of calculus gives

After integrating against |\phi_{ss}|ds, the error in replacing u(s)^2 by u(0)^2 is at most c\sqrt{\sup\epsilon}\,(u(0)^2+E_x), for \sup\epsilon\le1. The density and mean-curvature errors cost at most c\sup\epsilon\,(u(0)^2+E_x). Thus the form contribution of Equation (E.5) converges to 2\int H^{\rm old}_0u(0)^2, with the outward sign required in Lemma B.9.

The bounded strip terms have vanishing form cost because

The metric and density errors in the gradient and fixed bounded potential terms tend to zero in H^1 form norm. Integrate these estimates in x and use Equation (E.4); the stated error bound follows. One first proves it for smooth matching tests and then for the matching H^1 completion by density.

Choose \zeta on new hit packets, from deeper faces upward, after propagating the two incident-side requests through their finite diagrams. The function is an exact pullback in old product ends, so the displayed smoothing is also an exact product there. It is unchanged far out in the fresh cylinder coordinate. Since the normal coordinate and old metric already split off every old normal factor, the operation commutes with all the inherited collar identifications.□

Proof. Choose a locally finite coordinate cover, relatively compact smaller charts, and a smooth partition of unity \{\psi_\alpha\} subordinate to the smaller charts. Extend each chart slightly and average the function against a smooth nonnegative kernel of integral one, at a sufficiently small chart-dependent scale. Call the resulting local function T_\alpha f. Uniform continuity on the chart closure makes |T_\alpha f-f| smaller than the prescribed error on \supp\psi_\alpha. Thus

is a smooth approximation with that error. Every local average sends c to c, and \sum\psi_\alpha=1, so Tc=c exactly. Locally prescribed derivative accuracy is obtained in the same way wherever f is already smooth, using a slightly larger regular buffer.

Now use the finite tolerance allocation of Lemma A.4, and blend from deeper faces upward with the weights w_i in the proof of Lemma E.2. Their sum is one. Consequently blending preserves a common constant as well as exact face identities. The finite exceptional union over the prescribed directed dependencies is chosen before applying this construction. For every other entry the entire local input and all participating face copies coincide with the same constant, so all its averaging and blending errors are zero. No minimum of tolerances over these constant entries is needed. On the initial radial space the construction is local on buffered compact blocks and uses a locally finite chart cover. It has the same constant-reproduction property there.□

Proof. We justify restriction to a hypersurface, which does not follow merely by restricting an ambient almost-everywhere inequality. If f is locally Lipschitz on an open Riemannian set and |df|\le F almost everywhere for continuous F, then at every point its local Lipschitz slope is at most F. In a small coordinate ball, convolve f on a slightly larger ball. The derivative bound of the convolutions is at most the supremum of F there times a metric distortion factor tending to 1 as the ball shrinks. Integrating this bound along straight segments, passing to the uniform limit, and shrinking the ball proves the assertion. It follows also that every rectifiable curve satisfies the corresponding integrated bound, by subdividing into such balls.

Apply this on each open finite band of a profile, with F=(a_r\mu_j^2+\widetilde\Pi_j)/2. Restriction to the wall, inclusion of bulk, and projection on a cylinder therefore preserve the bound everywhere in local slope, and hence almost everywhere for the restricted locally Lipschitz function. Finite truncations remain locally Lipschitz, including where a pole is present. Across a seam the same conclusion follows by splitting a path into its bulk and cylinder portions; traces agree. A sufficiently small C^0 metric change increases all these slopes by at most 1+\eta_r.

For each label j, the raw potential has its own positive lower bound p_j^*>0. Apply Lemma E.5 to (1+3\eta_r/2)\widetilde\Pi_j, with strict value error smaller than \eta_r p_j^*/2. This proves the displayed relative bounds. Different labels may require different accuracies; a common positive lower bound for an infinite label list is unnecessary. When the raw potential is a spatial constant c_j on the whole relevant incident diagram, choose its new value to be exactly (1+3\eta_r/2)c_j. By Equation (E.3),

Continuous projections preserve the inclusion of every core closure inside its pole interior and preserve all prescribed q constraints.

Choose the seam form error small relative to the reserves of Lemma E.3. The potential in Equation (E.6) is at most that in Equation (E.1); the extra \eta_r supplies additional margin for the potential smoothing. Metric equivalence allows the required-set neighborhood to be slightly reduced after smoothing. Thus one may retain, for example, an e_0\|u\|_2^2 lower bound for the form of L_A^0 on tests in that neighborhood. No compatibility of the neighborhoods themselves is asserted.□

Proof. First disregard compatibility on a fixed packet. Put W_A=R_{\rm old}-V_A^*. Choose a smooth angle \theta equal to 0 near D_A and \pi/2 outside the neighborhood of local positivity, with bounded derivatives and with the support of \cos\theta inside that neighborhood. Uniform charts and the positive neighborhood width give such a function by smoothing a distance cutoff with buffers. Set

The identity for the squared partition \cos\theta,\sin\theta shows that Q_A^0+K_{\rm ind} has gap \gamma_A=\min\{e_0,J+\inf W_A\}>0. Indeed the gradient partition error is exactly C|\nabla\theta|^2u^2; local positivity controls the cosine part and W_A+J>0 controls the sine part. The correction is zero on a smaller uniform neighborhood of D_A.

Proceed from deeper faces upward. In sufficiently far elementary collars the already constructed K_B copies agree on intersections: on a common quadrant each is the pullback of the same further-face potential. They consequently define a single nonnegative function K_{\rm end} on the collar union. Copies are not added together on an intersection. The metric, measure, and V_A^* are likewise exact products there. In particular V^* has the same formula on a face: C, \lambda_{r+1}, b_{r+1}, the label set, and the old torus list do not depend on base dimension, after padding unused flat factors on each finite compared diagram.

For the finitely many elementary collars choose angles \theta_i which rise from 0 to \pi/2 over a normal interval of length L, inside the valid common product ranges. Use the squared partition

It has \sum_i\eta_i^2=1 and \sum_i|d\eta_i|^2\le\sum_i|d\theta_i|^2\le c_A L^{-2}; the inequality follows successively from the two-term sine/cosine identity, so no number of intersections is being summed separately. The support of \eta_0 lies in a compact collar truncation.

Choose smooth multipliers \alpha,\beta\in[0,1] with \alpha=1 on \supp\eta_0 and \beta=1 on the union of the face-cutoff supports. Let \alpha vanish only farther out than those transition intervals. Let \beta vanish on the inner side of a buffered collar union, so \beta K_{\rm end} extends smoothly by zero. Both can be formed from products of one-coordinate cutoffs. Define

On the central support this dominates K_{\rm ind}; on each collar support it dominates the single face copy. Eventually \alpha=0,\beta=1, giving exact compatibility. Nonnegativity permits the harmless overlap of these two terms in the finite transition region.

This correction vanishes in a uniform neighborhood of D_A. The individual correction already does. For the end term, a point of D_A in a product collar projects into D_B, because a face has a smaller required counting threshold. Start the support of \beta with a buffer from the collar entry. A point sufficiently close to D_A in this support then has a nearby required point in the same collar, and their face projections are just as close. It lies in the vanishing neighborhood of K_B. Take the minimum of the finitely many face widths and entry buffers; at corners the copies agree, so the same conclusion applies.

Apply the individual gap to \eta_0u. Apply each face gap by slicing to \eta_iu; its support is away from the collar entry, and the normal gradient term is nonnegative. If g_* =\min(\gamma_A,\{g_B\})>0, the finite IMS identity gives

Choose L so that the error is at most g_*/2. All quantities used for this choice are fixed before solving any elliptic equation. Boundedness and the derivative assertions follow from compact parts and finitely many product copies. Empty required sets or absent faces cause no change: one may take \theta=\pi/2 or omit the corresponding cutoff. This completes the compatible induction and proves the operator identity.□

Proof. Use the bounded geometry, positive bounded weight, and center-uniform polynomial volume estimate of Lemma A.2. The form gap and boundedness of W make the closed form coercive on H^1: combine Q\ge g\|u\|_2^2 with Q\ge C\|\nabla u\|_2^2-\|W_-\|_\infty\|u\|_2^2. It therefore has a self-adjoint inverse on L^2, of norm at most g^{-1}. If f\ge0, testing Lu=f with the negative part gives Q(u_-)\le0, so the inverse is positive.

Here is the decay estimate needed for bounded sources. We use an Agmon-type exponential-weight argument [1], proved here for the weighted packet operator. If f is supported in a metric unit ball I and u=L^{-1}f, put \chi=\exp(\delta\min\{d(\cdot,I),D\}), where C\delta^2<g. The form identity for a bounded Lipschitz multiplier gives

It holds by compact approximation in H^1; on the support of f, \chi=1. Hence

Let D\to\infty. The polynomial volume bound, uniform in the center of I, implies

Indeed sum the exponential bound on successive unit annuli against their polynomial volume upper bound.

Choose smooth compactly supported 0\le f_R\le1 equal to 1 on an increasing compact exhaustion, and put u_R=L^{-1}f_R\ge0. These are initially H^1 weak solutions. For each fixed R, the coefficients and f_R are smooth, and weak interior regularity makes u_R smooth [37, Lecture 6, Theorem 1 and Corollary 1]. This assertion requires no derivative bounds for f_R uniform in R.

For any h\in L^2(I), self-adjointness, Equation (E.8), and Cauchy–Schwarz with opposite exponential weights give

Thus u_R has a uniform local L^2 bound on every unit ball, with constant independent of its center and of R. Choose the uniform chart radius small enough that each larger chart ball lies in a metric unit ball. Writing B_1 and B_{1/2} for the fixed rescaling of two such nested chart balls, interior local boundedness gives

Here we use Simon [37, Lecture 18, Theorem 2, with p=2]; the fixed radius and coordinate changes are absorbed in c_A. The principal coefficients are uniformly elliptic and bounded, the lower-order coefficients are bounded, and the weight is bounded above and below. Since 0\le f_R\le1, this gives a bound independent of the center and R.

Fix two nested relatively compact coordinate domains. For all sufficiently large R, the source is exactly 1 on the larger domain. The interior Schauder estimate [37, Lecture 12, Theorem 2], followed by differentiation, therefore bounds every derivative of u_R on the smaller domain, uniformly for these R. A diagonal subsequence converges smoothly on compact subsets to a bounded nonnegative solution s of Ls=1. Applying the same estimates to this limiting equation in the uniform packet charts gives bounded derivatives of every fixed order on A.

The maximum argument must allow W to have either sign. Let w be a bounded smooth supersolution, put w_-=\max\{-w,0\}, and fix 0<\delta<\sqrt{g/C}. Write \psi=e^{-\delta d(x,x_0)} and take compact Lipschitz cutoffs \zeta_R equal to 1 on B(x_0,R), supported in B(x_0,2R), with |d\zeta_R|\le 2/R. Testing the supersolution with (\psi\zeta_R)^2w_-, and expanding the gradient, gives directly

All tests are compact, so no prior global gradient bound on w is needed. Boundedness of w and polynomial volume growth make \psi w_- square integrable. The terms containing d\zeta_R tend to zero, including the cross term by Cauchy–Schwarz. Dominated convergence therefore yields

so w_-=0. On an open subdomain the same compact-test argument applies when the negative part has zero boundary trace, using the Dirichlet form gap. In particular it applies to nonnegative boundary values. Apply comparison to both signs of the difference of two bounded solutions to obtain uniqueness. If M=1+\|W\|_\infty, then L(s-M^{-1})=1-W/M\ge0, and comparison gives s\ge M^{-1}.

Euclidean products retain bounded geometry and polynomial volume, and their form gap follows by slicing. The preceding proof therefore applies verbatim; uniqueness identifies their solution with the pullback of s.□

Proof. We first establish a uniform end estimate over a whole face, which also avoids any compactness ambiguity at its deeper ends. On an exact elementary collar B\times[T,\infty), the difference w=s_A-s_B is bounded and satisfies L_Aw=0. Set m_B=\inf s_B>0 and choose a>0 with Ca^2\|s_B\|_\infty\le1/2. The function

satisfies L_Ab=e^{-a(t-T)}(1-Ca^2s_B)\ge\tfrac12e^{-a(t-T)}. Comparison on the half collar, using its sliced Dirichlet gap and the Dirichlet comparison argument in the proof of Lemma E.8, yields

The boundary inequalities hold at t=T for both signs. Uniform interior elliptic estimates for the homogeneous difference give the same exponential decay in C^2 on t\ge T+1. All constants are uniform in x\in B, including its deeper ends. Thus the estimate holds on each elementary collar, also where collars overlap. Exact agreement of the approximations on those overlaps will come from the compatible face copies used in the following construction.

All operator coefficients and every exact s_A are now fixed. Let M_A be a finite bound for L_A:C^2\to C^0 in uniform charts, and choose a target

The finite family of collar blending cutoffs has a C^2 operator bound H_A\ge1, independent of translation of their starts. Include in H_A the fixed finite factor for the number of copies. For each packet B, let \varepsilon_B be the minimum of

over all directed coface-to-face chains ending at B, including the length-zero chain. There are finitely many such chains, so this minimum is positive. Appending the edge A>B to a minimizing chain for A gives \varepsilon_B\le\varepsilon_A/(4H_A). All these numbers are fixed after the corrections and operator norms, and before constructing any v_A.

Construct v_A from deeper faces upward. Keep s_A in the central part and use the ordinary weights in the proof of Lemma E.2 to blend to the already constructed copies of v_B. Move the starts until \|s_A-s_B\|_{C^2}<\varepsilon_A/(4H_A) on all participating collars, using Equation (E.10). The persistent difference v_B-s_B is paid by its previously allocated tolerance, not by moving the start. Start also beyond the finitely many compatibility thresholds for the v_B’s, so their copies agree on overlaps. The derivative bounds for the weights then give

The result is an exact face copy sufficiently far out, is smooth, and has bounded higher derivatives by compact parts and finitely many smooth products. Finally v_A\ge\tfrac34\inf s_A>0 and L_Av_A\ge1-M_A\tau_A\ge3/4, which implies Equation (E.9). No geometry radius, correction, or operator is changed during this approximation.□

Proof. Use Lemmas E.2, E.4 and E.6 for the smooth base metrics and the profile data, and Lemmas E.7 and E.9 for v_A. Append k=399 circle variables, each with length function v_A^{2/(k+1)}. The warped-product identity is

Indeed a common scale f contributes -2k f^{-1}\lap_{\rm old}f-k(k-1)f^{-2}|df|^2; substitution of f=v_A^{2/(k+1)} cancels the separate |dv_A|^2 term. The old Laplacian on invariant functions is the weighted base Laplacian, including the product of all old warp lengths. Here k counts only the newly appended circle factors. Any unit factors used earlier to compare old lists on a finite diagram have zero logarithmic derivative and leave both \lap_{\rm old} and this coefficient unchanged. Since 4k/(k+1)=C, Equations (E.6) and (E.9) give

Here K_A=0 on a neighborhood of D_A. Adding these factors does not change base distances or profile gradients. The new factors are positive, bounded above and away from zero, and exactly compatible. The same number is appended in every status at the stage, so all end identities compare the same list of factors. This proves the analytic invariant, including when the new required threshold is zero. The componentwise metric operations preserve every unused old circle length identically equal to 1: restriction, product extension, and \beta_s\mapsto\beta_{\phi(x,s)} all fix that constant entry. The 399 appended lengths are the only common new entries. Likewise restriction and cylinder pullback preserve a constant pole profile and its full core, and Lemma E.5 preserves the specified multiplied constant potentials.

The zero wall lies in \operatorname{int}G_{j_0} and all current forced signs hold by Lemma E.2; earlier forced cores are carried by the continuous projection and retain their pole neighborhoods. Thus this is exactly the regular compatible splitting operation required by the packet construction. At the radial split, the preserved alternating compact barrier bands separate all wall components into bounded transition shells and trap every closed sign-side component between two separating bands (with a compact initial block). These components are compact before prolongation. The normal collars and smooth metrics used above are products in all old ends. Lemmas A.1 and A.2 therefore supply the claimed geometry after prolongation. The main argument applies the one-dimensional obstruction and reconstructs the compatible partitions in the original manifold.□