Proof. The assertion is immediate when \(\|M\|=0\), so assume that \(\|M\|>0\). Write \(Y(M,[g])\) for the Yamabe constant of the conformal class of \(g\): \[Y(M,[g])= \inf_{u\in C^\infty(M),\,u>0} \frac{\displaystyle\int_M \left(\frac{4(n-1)}{n-2}|du|_g^2+\mathop{\mathrm{Scal}}_g u^2\right)dV_g} {\displaystyle\left(\int_Mu^{2n/(n-2)}\,dV_g\right)^{(n-2)/n}}.\] The Yamabe theorem, developed through the work of Yamabe, Trudinger, Aubin, and Schoen [25, 24, 2, 23], gives a conformal metric attaining this value and having constant scalar curvature; see also [15]. Its scalar curvature is negative by Theorem 2. After a constant rescaling, it becomes a metric \(h\) with \(\mathop{\mathrm{Scal}}_h=-1\). Evaluating the Yamabe quotient at the constant function for this minimizing metric gives \[-Y(M,[g])=\mathop{\mathrm{Vol}}_h(M)^{2/n}.\] For each positive \(u\), Hölder’s inequality gives \[\int_M\mathop{\mathrm{Scal}}_g u^2\,dV_g \ge -\left(\int_M(\mathop{\mathrm{Scal}}_g^-)^{n/2}\,dV_g\right)^{2/n} \left(\int_Mu^{2n/(n-2)}\,dV_g\right)^{(n-2)/n}.\] The gradient term in the Yamabe quotient is nonnegative. Taking the infimum therefore yields \[-Y(M,[g]) \le \left(\int_M(\mathop{\mathrm{Scal}}_g^-)^{n/2}\,dV_g\right)^{2/n}.\] Consequently \(\mathop{\mathrm{Vol}}_h(M)\le\int_M(\mathop{\mathrm{Scal}}_g^-)^{n/2}\,dV_g\), and (2) implies (3). ◻

Proof. Write \(\epsilon(v)=\sum_\ell v_\ell\) for the mass of a finitely supported vector. Multilinear extension of the cocycle equation gives \[0=\sum_{j=0}^{n+1}(-1)^j\epsilon(w_j) c(w_0,\ldots,\widehat w_j,\ldots,w_{n+1}).\] Choose \(w_0\) of mass one and let \(w_1,\ldots,w_{n+1}\) have mass zero. It follows that \(c(w_1,\ldots,w_{n+1})=0\). All tangent vectors to a probability simplex have mass zero. Differentiating (6) now shows \(d\eta=0\).

To construct \(F\), choose finitely many evenly covered open sets in \(M\) and smooth nonnegative bumps subordinate to them whose nonzero sets cover \(M\). Lift each bump to one chosen sheet and take all its deck translates, denoting the resulting functions by \(b_\ell\). Their supports are compact and locally finite on \(\widetilde M\). Set \[F_\ell=\frac{b_\ell}{\left(\sum_j b_j^2\right)^{1/2}}.\] The denominator is positive, and the stated properties follow.

Choose a finite oriented triangulation of \(M\). On its lift, assign one label to every vertex equivariantly, and let \(p^{\mathrm{aff}}\) be the probability map affine in barycentric coordinates on each simplex. The homotopy \((1-t)p+t p^{\mathrm{aff}}\) is equivariant and has locally finite support. Applying Stokes’ theorem to each simplex times \([0,1]\), and cancelling the common faces, shows that its pullback integral is constant. On a simplex with ordered vertex labels \(\ell_0,\ldots,\ell_n\), write \[p^{\mathrm{aff}}=e_{\ell_0} +\sum_{j=1}^nt_j(e_{\ell_j}-e_{\ell_0}).\] Alternation makes the value of \(c(p^{\mathrm{aff}},dp^{\mathrm{aff}},\ldots,dp^{\mathrm{aff}})\) on the coordinate frame constant and equal to \(c(\ell_0,\ldots,\ell_n)\). The coordinate simplex has volume \(1/n!\), so the factor \(n!\) in (6) makes its integral exactly this cocycle value. Summing over the oriented fundamental triangulation gives (7). The identification with the chosen group class is the usual cochain realization of the Mapping Theorem: an equivariant vertex labeling on the universal cover evaluates the homogeneous group cochain, and two such labelings give cochain-homotopic maps by the simplex prism construction. ◻

Proof. Write \(\bar h=h+dz^2\), and suppress pullback by the graph projection. The formula for the graph mean curvature gives \(H=-\mathop{\mathrm{div}}U\), while \(\partial_\nu\phi=-U(\phi)\); hence the stated formula for \(D\) holds. The rank-one determinant formula gives \(dV_{h'}=L\,dV_h\), proving the first assertion in (10). The inverse rank-one formula gives its second assertion.

Let \[P=|\mathrm{II}|^2+ \bigl(\mathop{\mathrm{Ric}}_{\bar h}-\bar\nabla^2\phi\bigr)(\nu,\nu).\] The Gauss equation and the decompositions of the Laplacian and gradient of \(\phi\) give \[\begin{align*} \mathop{\mathrm{Scal}}_{h'}&=\mathop{\mathrm{Scal}}_{\bar h}-2\mathop{\mathrm{Ric}}_{\bar h}(\nu,\nu) +H^2-|\mathrm{II}|^2,\\ \Delta_{\bar h}\phi &=\Delta_\Sigma(\phi|_\Sigma) +H\partial_\nu\phi+\bar\nabla^2\phi(\nu,\nu),\\ |d(\phi|_\Sigma)|^2&=|d\phi|_{\bar h}^2-(\partial_\nu\phi)^2. \end{align*}\] Since \(\mathcal S(\bar h,\phi)=\mathcal S(h,\phi)\), their combination is \[ \mathcal S(h',\phi|_\Sigma) =\mathcal S(h,\phi)+D^2-2P+|\mathrm{II}|^2. \tag{11}\] Adding \(\log v\) to the restricted weight changes the left side by \[-2\Delta_{\Sigma,\phi}\log v-|d\log v|^2 =-\frac{2\Delta_{\Sigma,\phi}v}{v}+|d\log v|^2 .\]

The tangential part of vertical translation is \[T=\partial_z-v\nu=(vU,1-v^2).\] For a hypersurface variation with normal speed \(v\), the variation of \(H\) is \(-\Delta_\Sigma v-(|\mathrm{II}|^2+\mathop{\mathrm{Ric}}_{\bar h}(\nu,\nu))v\). The additional variation of \(\partial_\nu\phi\) is \[v\bar\nabla^2\phi(\nu,\nu) -\langle d(\phi|_\Sigma),dv\rangle_\Sigma .\] Including the tangential velocity \(T\), and using that vertical translation preserves the metric, weight, and \(D\), therefore gives \[ (\Delta_{\Sigma,\phi}+P)v=T(D). \tag{12}\] The horizontal part of \(T\) is \(vU\), so \(T(D)=vU(D)\). Substitution in (11) proves (9). Finally, differentiating \(v=\langle\nu,\partial_z\rangle\) tangentially gives \(dv(V)=\mathrm{II}(V,T)\). Since \(|T|^2=1-v^2=|U|^2\), it follows that \(|d\log v|\le v^{-1}|U||\mathrm{II}|\), which is the last bound in (10). ◻

Proof. We give existence and the estimates together, since the estimates also give uniqueness and measurable dependence.

Height barriers and comparison. In Case (i), replace \(F_i,\beta\) by \(tF_i,t\beta\), \(0\le t\le1\). At \(t=0\) the constant \(f=\delta\) is a solution. For every \(t\), the constant graphs \(f=c_{\min}\) and \(f=2\) are a lower and an upper barrier, respectively. Indeed their weighted mean curvature is zero, and at angle \(v=1\) the right side of (18) is nonnegative at \(f=c_{\min}\) and strictly negative at \(f=2\). In Case (ii), replace \(f_0\) by \(tf_0\), starting with \(z=0\). For \(\tau=\pi/2-\eta\), the graphs \[z_-=\frac{tf_0-w\tau}{e},\qquad z_+=\frac{tf_0+w\tau}{e}\] have arguments \(+\tau\) and \(-\tau\), respectively. Their weighted mean curvatures have a common bound independent of \(\eta\), because their derivatives do not involve \(\tau\). Choose \(\eta>0\) so small that \(B_{\min}\tan\tau\) exceeds this bound. These are then lower and upper barriers for every \(t\). They give the asserted height and pole margins.

Both comparisons use strict decrease of \(p\) in its height argument, with the other arguments held fixed: \[\begin{align*} p_z&=-\frac{\epsilon a(F_i^2+\delta^2)+(1-v^2)/2}{z^2} -\frac{\epsilon a+\beta}{N^2} \le-\frac{\epsilon a}{N^2} &&\text{in Case \textup{(i)}}, \\ p_z&=-\frac{eB}{w}\sec^2\bigl((f_0-ez)/w\bigr) \le-\frac{eB_{\min}}w &&\text{in Case \textup{(ii)}}. \tag{21}\end{align*}\] At an interior positive maximum of the difference between a solution and an upper barrier, their gradients, hence their angles, agree. The negative elliptic principal part of \(D_z\) gives a nonnegative difference of mean curvatures there, whereas strict height decrease gives a negative difference of prescriptions. This is impossible. The lower comparison and uniqueness follow in the same way. On a noncompact manifold the maximum can be replaced by a supremum: take uniform charts centered at an extremizing sequence, subtract the value of \(\phi\) at each center, and pass to a local limit. For any individual solution in the stated uniqueness class, its uniform \(C^{2,\alpha}\) bound and the coefficient bounds give such a limit, which attains the extremum at the center of an interior chart. This justifies the same comparison without compactness.

The angle bound. For a solution, (12) and the chain rule give \[ (\Delta_{\Sigma,\phi}+P)v_z =v_z\langle p_x,U_z\rangle +p_z(1-v_z^2)+p_vT(v_z). \tag{22}\] Here \(p_x\) differentiates with \(z,v\) fixed, and we used \(T(z)=1-v_z^2\). The ambient curvature and weight bounds imply \(P\ge-C\), since the remaining term \(|\mathrm{II}|^2\) is nonnegative. At an infimum of \(v_z\), taken by the same recentering procedure, \(\Delta_{\Sigma,\phi}v_z\ge0\) and \(T(v_z)=0\). Consequently, if \(v_z\le1/2\) there, \[ (1-v_z^2)|p_z|\le v_z(C+|p_x|). \tag{23}\] The height margins just proved bound \(p_x\) uniformly, and (21) bounds \(|p_z|\) below by a positive constant. Since \(1-v_z^2\ge3/4\) in this case, (23) gives a uniform positive lower bound for \(v_z\). In using recentered limits at this stage, a solution is first in an individual uniform \(C^{2,\alpha}\) class. Ordinary local bootstrapping gives its individual higher regularity before the common estimates are deduced, so the differentiations and limits in this argument are justified.

Higher estimates. In a uniform chart let \(\omega=e^\phi\sqrt{\det h}\), and divide \(\omega\) by its value at the chart center. The resulting density has a common upper bound and a positive lower bound, and all its derivatives are bounded. The equation is of the form \[ -\partial_j A^j(x,\partial z)=F(x,z,\partial z). \tag{24}\] For example \(A^j=\omega h^{jk}z_k/L_z\) after the normalization. On the established height and gradient ranges the flux and the right side are smooth with uniformly bounded derivatives, and \(\partial A^j/\partial z_k\) is uniformly elliptic. Differentiating (24) gives linear divergence equations for \(\partial_kz\). Their leading coefficients are bounded and uniformly elliptic. Gradient dependence of \(F\) produces a bounded first-order coefficient; the remaining terms are a bounded function plus the divergence of a bounded vector field. The term \(F_z\partial_kz\) is included in the bounded source, since the gradient is already bounded.

The interior De Giorgi–Nash estimate for these equations, with bounded lower-order coefficients and source data in \(L^{s/2}\) and \(L^s\) for \(s>\dim X\), gives a common Hölder bound for \(\partial z\); see [9]. The coefficients in the nondivergence form of the original equation are now Hölder continuous. Interior Schauder estimates give a uniform \(C^{2,\alpha}\) bound for some \(\alpha>0\), after shrinking within the chart buffers. Differentiating and applying Schauder estimates again yields uniform bounds for every derivative. All these estimates are local with common buffers, so their constants are uniform throughout \(X\).

Continuation. Work in uniform \(C^{2,\alpha}\) spaces on the open set of functions with a uniform height margin. The equation \(D_z-p(x,z,v_z)=0\) defines a smooth map to uniform \(C^{0,\alpha}\). At a solution its linearization has negative uniformly elliptic principal part, uniform Hölder coefficients, and zeroth-order coefficient \(-p_z\) bounded below by a positive constant. This linear operator is invertible on the uniform spaces. To see surjectivity, exhaust a noncompact component by smooth relatively compact domains and solve the linear Dirichlet problem with zero boundary data on each domain. The maximum principle bounds the supremum of the solutions by the supremum of the right side divided by the positive zeroth-order lower bound. Interior Schauder estimates then give the same \(C^{2,\alpha}\) bound in every buffered chart once it lies in the exhaustion. A diagonal local limit solves the equation on the component with the uniform bound. On a compact component the corresponding closed-manifold elliptic theory gives the same conclusion. Uniqueness follows from the maximum principle, using recentered limits on noncompact components. The estimate also makes the inverse bounded.

The inverse function theorem gives openness of the set of homotopy parameters with a solution. The a priori height, angle, and higher bounds are common over the homotopy. For a sequence of parameters converging to a limit, diagonal smooth convergence on a countable chart cover therefore gives a solution at that limit with the same bounds and height margin. This gives closedness. Each homotopy starts from the constant solution specified above, so existence follows. Comparison gives the claimed uniqueness.

The graph estimates now follow from (10). In particular \(h_z\ge h\), and the gradient bound also gives a uniform upper comparison in the given charts. Thus \(h_z\) is complete and has uniform buffered charts with bounds on all derivatives. The corresponding derivative bounds on \(\log L_z\) give those for \(\phi_z\). This also permits successive applications of the lemma.

Borel dependence. For completeness, fix a starting plaque and enumerate the charts accessible along its leaf by words in the Borel overlap maps. Represent a solution by its functions in this countable family of charts, in a countable product of smooth function spaces. Compatibility on overlaps, the equation, the height margin, and the common derivative bounds are Borel conditions: each can be tested on countably many compact subcharts and dense sets, where the relevant functions and their derivatives are continuous. The chart data and changes of coordinates are Borel by hypothesis. The resulting Borel relation between data and local representatives has a singleton fiber over each set of data, by existence and uniqueness. The Borel injection theorem [14], applied to its projection to the data, makes the inverse projection Borel. This proves measurable dependence. The coding refers to a full prescribed problem on the same complete leaf. It applies independently of the transverse mass of the chart family. Finally, applying an equivariant identification to a solution produces a solution of the identified problem, so uniqueness proves equivariance. ◻

Proof. We first construct the outer coefficient while keeping its transition width bounded. Starting at a fixed positive distance from \(K\), a scalar profile \(b\) can increase from zero to \(H_*\) at speed at most \[\frac{c_0}{C N}\bigl(\theta\epsilon^{3/2}+b^{3/2}\bigr),\] where the fixed factor \(C\) leaves room for the estimates below. Choose the speed comparable to this expression during the increase, with smooth flattening at either end. The width needed is at most \[ C N\int_0^\infty\frac{dt} {\theta\epsilon^{3/2}+t^{3/2}} =C\frac{N}{\theta^{1/3}\sqrt\epsilon} =C\frac{\sqrt l}{\theta^{1/3}}\le C . \tag{29}\] For example one can use the primitive of the reciprocal of a smooth positive function comparable to \(\theta\epsilon^{3/2}+t^{3/2}\) as the new profile coordinate, and compose with fixed smooth flattenings. The last inequality uses \(l\le\theta\le1/16\).

One may use this profile in a smooth approximation of \(\mathop{\mathrm{dist}}_{h_{\mathrm{ref}}}(x,K)\) with gradient at most \(2\) and a uniformly small error. Here is a construction with the needed uniformity. Shrinking the controlled charts gives normal charts at a fixed positive radius, by the uniform geodesic equation estimates. Average the distance function in such a chart centered at \(x\) against a fixed smooth radial kernel in \(T_xX\), at a sufficiently small radius. Written as an integral against a kernel in \((x,y)\), this is a smooth function of \(x\). The error is at most the averaging radius. To bound the first derivative, move the center and parallel transport the averaging vectors. At a sufficiently small radius the displacement of the exponential images is at most twice the displacement of the center; the distance function is one-Lipschitz. The derivative bound follows. Higher derivatives have uniform bounds from the smooth local kernels: in differentiating the normalized kernel one may subtract the distance at the center, leaving a uniformly bounded oscillation on its support. These derivative bounds hold in the common buffered charts, since the \(h_{\mathrm{ref}}\) exponential maps and kernels have uniformly bounded derivatives there. The inverse metric inequality \(h^{-1}\le h_{\mathrm{ref}}^{-1}\) then bounds the gradient of the composed profile in \(h\).

Choose this raw profile \(b\) to vanish near \(K\), to equal \(H_*\) past the width in (29), and to leave fixed outer buffers. To accommodate \(\alpha\), take a smooth nondecreasing saturation \(\sigma_{H_*}:[0,\infty)\to[0,H_*]\) that equals the identity up to \(H_*/3\), equals \(H_*\) from \(H_*\) on, and has a derivative bounded by a numerical constant. Put \[\beta=\sigma_{H_*}(b+\alpha),\qquad \beta_s=\sigma_{H_*}(b+\alpha)-\alpha .\] Take \(H_*\) much larger than the value and derivative bounds of the already fixed \(\alpha\). Then \(\beta_s\ge0\), and it equals \(b\) near \(K\). Wherever the saturation differs from the identity, both \(\beta_s\) and the scale of \(b\) are comparable to \(H_*\). The formula \[d\beta_s=\sigma_{H_*}'(b+\alpha)\,db +\bigl(\sigma_{H_*}'(b+\alpha)-1\bigr)d\alpha\] therefore proves (25): choose the initial fixed speed factor large enough to handle the first term, and then choose \(H_*\) large enough for the fixed \(d\alpha\) term to be a small fraction of \(H_*^{3/2}/N\). Where saturation is the identity this is just the raw profile bound. Where \(b=H_*\), the total \(\beta\) equals \(H_*\). A cutoff supported beyond fixed additional buffers can consequently be chosen once at a radius \(R_{\mathrm{loc}}\) as in the statement. Take it to be the square of a smooth nonnegative cutoff that is flat at its outer end. This gives \(|d\chi|^2\le C\chi\) and the other asserted cutoff properties. If some outer regions are empty, the corresponding assertions are vacuous.

We next prove that the complete solution can be cut on this plateau. All constants in the rest of the proof may use the fixed data and parameters, but not \(H_*\). Write \[c_H=H_*^{-2}\sqrt{\frac{\epsilon a}{\epsilon a+H_*}} \asymp H_*^{-5/2}.\] By (20), \(c_H\le\widetilde f\le2\). Cover the cutoff region by inner chart boxes, each with a fixed succession of larger buffered boxes whose closures lie in the plateau \(F_i=0,\ \beta=H_*\). Their sizes may use the current \(h\)-chart bounds inside the fixed \(h_{\mathrm{ref}}\) buffers; all these bounds were fixed before \(H_*\), so the boxes and their buffers have sizes independent of \(H_*\). Our target is that, for every integer \(j\ge0\) and every \(M>0\), there is \(C_{j,M}\) such that, for all sufficiently large \(H_*\), \[ \|\widetilde f-c_H\|_{C^j} \le C_{j,M}H_*^{-M}, \tag{30}\] on the inner boxes, with norms in these fixed coordinates. The constants use only the indicated data and parameters fixed before \(H_*\). The family assertion of Lemma 6 is applied separately for each fixed \(H_*\); the estimates that follow establish this superpolynomial decay uniformly as \(H_*\) grows.

There is first a global angle bound \[ v\ge\frac{c_1}{1+\sqrt{H_*}} \tag{31}\] for the uncut graph. In fact, at a recentered infimum with \(v\le1/2\), the formulas for \(p_x,p_z\) and (25) give \[|p_z|\ge c_2(1+\beta+\widetilde f^{-2}),\qquad |p_x|\le C\bigl(\widetilde f^{-1} +\widetilde f(1+\beta^{3/2})\bigr).\] For the first bound use \(1-v^2\ge3/4\) in \(p_z\); for the second use the fixed bounds on \(F_i,\alpha\) and their derivatives. Since \(\widetilde f\le2\) and \(\beta\le H_*\), the ratio of the right side of the second estimate, plus a constant, to the right side of the first is at most \(C(1+\sqrt{H_*})\). Equation (23) proves (31). This recentering is done for each fixed \(H_*\), using the individual regularity supplied by Lemma 6.

On each of the outer plateau boxes the equation is \[ ND=\frac{\epsilon a\delta^2}{\widetilde f} +\frac{1-v^2}{2\widetilde f} -(\epsilon a+H_*)\widetilde f, \qquad (\epsilon a+H_*)c_H^2=\epsilon a\delta^2 . \tag{32}\] There is an exponential value estimate on an inner box: \[ 0\le\widetilde f-c_H\le C\exp(-c_3\sqrt{H_*}). \tag{33}\] To prove it, let \(d_{\mathrm{face}}\) be ordinary coordinate distance to a face of the outer box and put \[w_1=\sum_{\mathrm{faces}}3 \exp(-\gamma\sqrt{H_*}\,d_{\mathrm{face}}).\] On the boundary, \(c_H+w_1\ge3\ge\widetilde f\). Its first and second coordinate derivatives are bounded by \(C\gamma\sqrt{H_*}w_1\) and \(C\gamma^2H_*w_1\), respectively. Direct differentiation of the weighted graph divergence therefore bounds its mean curvature from below by \[-C(\gamma^2H_*+\gamma\sqrt{H_*})w_1 .\] For this graph, the height terms in its prescription satisfy \[\frac{\epsilon a\delta^2}{N(c_H+w_1)} -\frac{(\epsilon a+H_*)(c_H+w_1)}N =-\frac{\epsilon a+H_*}{N}w_1 \left(1+\frac{c_H}{c_H+w_1}\right).\] The remaining angle term is at most \(C\gamma^2H_*w_1\), because \((1-v^2)/(c_H+w_1)\le N^2|dw_1|^2/(c_H+w_1)\le C\gamma^2H_*w_1\). Choosing \(\gamma>0\) sufficiently small and then \(H_*\) sufficiently large makes \(c_H+w_1\) an upper barrier. Comparison with this barrier and the lower constant \(c_H\) proves (33) on any inner box a fixed distance from the faces.

On a smaller box the angle is bounded below independently of \(H_*\). To see this, choose a smooth chart cutoff \(0\le\xi\le1\), supported in a region where (33) holds and equal to one on the smaller box. Minimize \(v/\xi\) over \(\{\xi>0\}\). The global lower bound (31) makes a minimum with value below \(1/2\), if one exists, interior, and at that minimum \[v<\xi/2,\qquad \xi\ge c/\sqrt{H_*},\qquad \nabla_\Sigma v=(v/\xi)\nabla_\Sigma\xi,\qquad \Delta_{\Sigma,\phi}v\ge(v/\xi)\Delta_{\Sigma,\phi}\xi.\] The formula for the restriction of an ambient function to the graph gives \(\Delta_{\Sigma,\phi}\xi\ge-C(1+|\mathrm{II}|)\). The positive \(|\mathrm{II}|^2\) term in \(P\), and completion of its square against \(C|\mathrm{II}|/\xi\), show that the left side of (22) is at least \(-CH_*v\). On this plateau \(p_x=0\). Since \(v<1/2\), the \(p_z\) term on its right side is at most \(-c/\widetilde f^2\). Also \(p_v=-v/(N\widetilde f)\) and \(|T(v)|\le Cv/\xi\), so that right side is at most \[-\frac{c}{\widetilde f^2} +\frac{C\sqrt{H_*}\,v}{\widetilde f}.\] Combining the two bounds and multiplying by \(\widetilde f^2\) would give \[c\le CH_*\widetilde f^2+ C\sqrt{H_*}\,\widetilde f .\] By (33), \(\widetilde f\le2c_H\) for large \(H_*\), and the last right side tends to zero. This contradiction proves \(v\ge1/2\) on the smaller box.

It remains to control derivatives on the cutoff region without using any growth rate for the global elliptic constants. In a still smaller box, rescale coordinates \(x=x_0+c_Hy\) and write \(\widetilde f=c_Hg\). The weighted graph curvature in the rescaled chart is \(c_HD\), and its right side becomes \[\begin{align*} c_Hp(x,Nc_Hg,v) &=\frac{\epsilon a\delta^2+(1-v^2)/2}{Ng} -\frac{c_H^2(\epsilon a+H_*)g}{N}\\ &=\frac{\epsilon a\delta^2+(1-v^2)/2}{Ng} -\frac{\epsilon a\delta^2g}{N}. \end{align*}\] Here \(1\le g\le2\) for large \(H_*\), and the angle bound gives a uniform gradient bound. The rescaled metric and normalized weight have uniformly bounded derivatives; the displayed right side and all its derivatives are uniformly bounded on these ranges. The local higher estimates in the proof of Lemma 6, applied on unit boxes in these coordinates, consequently give uniform bounds for every derivative of \(g\). Scaling back gives bounds of the form \(C_jc_H^{1-j}\) for the \(j\)-th derivatives of \(\widetilde f\). Interior interpolation between these polynomial bounds and the exponential bound (33), on further fixed buffered boxes, proves (30). Indeed the usual finite-difference interpolation of a \(j\)-th derivative between the supremum norm and a higher derivative norm gives an exponential factor times a power of \(c_H^{-1}\), which is smaller than every power of \(H_*^{-1}\).

We can now cut off. On the transition region, \(\widetilde f\asymp c_H\), its positive-order derivatives are smaller than every power of \(H_*^{-1}\), and the derivatives of \(\chi\) are fixed. On \(\{\chi>0\}\), \[\frac{|d(\chi\widetilde f)|^2}{\chi\widetilde f} \le 2\widetilde f\,\frac{|d\chi|^2}{\chi} +2\chi\,\frac{|d\widetilde f|^2}{\widetilde f} =O(c_H).\] It follows that \(A^2/f\le N^2|df|^2/f\) tends to zero. All fixed derivatives of \(f\) are \(O(c_H)\) there. The graph formulas therefore give \(ND,N|\mathrm{II}|\), and \(N|d\log L|\) tending to zero. They also control \(\nabla D\), so (9) gives the first limit in (28). Finally \(F_i=0\) in this region, \(ND^0=-\epsilon af+A^2/(2f)\), and \(\beta f\le H_*\,O(c_H)\to0\). This proves every limit in (28). To verify the extensions at the zero set, write \(\chi=\zeta^2\), where \(\zeta\) is smooth and flat there. On \(\{\zeta>0\}\), \[\frac{|df|_h^2}{f} =4\widetilde f\,|d\zeta|_h^2 +4\zeta\langle d\zeta,d\widetilde f\rangle_h +\zeta^2\frac{|d\widetilde f|_h^2}{\widetilde f}.\] The right side extends smoothly and flatly across \(\{\zeta=0\}\), since \(\widetilde f>0\). The same is therefore true of \[\frac{A^2}{f} =\frac{N^2}{1+N^2|df|_h^2}\,\frac{|df|_h^2}{f}.\] Moreover, \(F_i\) vanishes on a neighborhood of the zero set of \(f\). Thus \(ND^0\) extends smoothly by zero there, and all jets of \(f=\zeta^2\widetilde f\) vanish on its zero set.

On the uncut region the last term \(\varpi=\epsilon a\delta^2/f\) in (27) has \[ \|\varpi\|_\infty \le\delta\sqrt{\epsilon a(\epsilon a+H_*)}\longrightarrow0, \qquad -2NU(\varpi) =\frac{2\epsilon a\delta^2 L A^2}{f^2}\ge0 . \tag{34}\] Since \(ND^0-\beta f\ge-(\epsilon a+\beta)f\), its contribution inside the curvature square satisfies \[ (ND^0-\beta f+\varpi)^2 \ge (ND^0-\beta f)^2 -2\epsilon a\delta^2(\epsilon a+H_*). \tag{35}\] The last loss tends to zero. These estimates on the core and (28) on the transition prove the claimed bound for \(\mathrm{err}\). The curvature estimates use the derivative sign in (34) and the full transition estimates, as well as that bound on the value of \(\mathrm{err}\).

Every construction has uniform bounds for fixed \(H_*\). For Borel data, the distance to the open nonzero set of \(F_i\) is Borel: using the enumerated charts it is the infimum of lengths of a countable dense collection of chart paths ending where \(F_i\ne0\). The normalized smoothing kernels and all subsequent scalar operations are Borel and equivariant. Lemma 6 supplies these same properties for the complete solution. This proves the final assertions. ◻

Proof. We use sweeps with \(\phi_0=0\), no boost, and \(l=\theta\), where \(l>0\) will be a sufficiently small fixed dimensional number. Then \(\phi_+=-W_p\). After a sweep set \[k=e^{-2W_p/(n-1)}h_+ .\] The scalar conformal formula and the definition of \(\mathcal S\) give \[ \mathop{\mathrm{Scal}}_k=e^{2W_p/(n-1)} \left(\mathcal S(h_+,\phi_+)+\frac{|dW_p|_{h_+}^2}{n-1}\right). \tag{60}\] The same reset contracts the volume density: \[ dV_k=e^{-nW_p/(n-1)}dV_{h_+} =e^{-W_p/(n-1)}dV_{h_0}\leq dV_{h_0}. \tag{61}\] We claim that there are dimensional \(C_b\) and \(\epsilon_b>0\) such that, whenever \(2\leq C_0\leq4\), \(0<\epsilon\leq\epsilon_b\), and \(\mathop{\mathrm{Scal}}_{h_0}\geq-C_0+\epsilon e_{h_0}(F)\), one can choose the errors in the sweep so that \[ \mathop{\mathrm{Scal}}_k\geq-(C_0+C_b l\epsilon)+(1+l)\epsilon e_k(Y). \tag{62}\]

We first derive the budget needed for this step. The reset gives \(e_k(Y)=e^{2W_p/(n-1)}e_{h_+}(Y)\). Substituting the definition of \(Q\) and the input scalar inequality into (60), and dropping its nonnegative gradient term, shows that it suffices to have \[\frac{\epsilon Q}{rN^2}\geq l\epsilon e_{h_+}(Y)+C_0(1-e^{-2W_p/(n-1)}) -C_b l\epsilon e^{-2W_p/(n-1)}.\] Multiply by \(rN^2\), and use \(N^2 e_{h_+}(Y)\leq m/r\) and \(N^2=l\epsilon\). A sufficient budget for (62) is therefore \[ \epsilon Q\geq l\epsilon m+C_0l\epsilon r(1-e^{-2W_p/(n-1)}) -C_b l^2\epsilon^2r e^{-2W_p/(n-1)}. \tag{63}\]

To pay this budget when \(\epsilon\) is small, we need control of both \(m\) and the deviation of \(r\) from one. The sweep inequality implies \[ \epsilon Q\geq c_n\bigl(\epsilon^2(1-r)^2+m^2\bigr) -C_n\epsilon l^2m-r\delta'_0 , \tag{64}\] where \(\delta'_0>0\) can be chosen arbitrarily small. Here is a proof. Fix a sufficiently small dimensional \(m_b>0\). If \(m\geq m_b\) and \(\epsilon_b\) is small enough, then \(\bar u=m/\epsilon\geq4a\). Since \[-M_0=ar+\frac{\bar u}{2}-a,\] the square \(\epsilon^2M_0^2\) controls \(\epsilon^2(1-r)^2+m^2\). This is one of the positive terms in \(\epsilon\Pi_p\).

Suppose next that \(m<m_b\). The prescription term \(D_i^0\) of Proposition 8 satisfies \[ND_i^0+t_i=\frac{\epsilon f_i}{r} \left[V_i+M_0+(\lambda+\tfrac12)\bar u+ (\tfrac32-\lambda)u_i\right].\] Both displayed coefficients of \(\bar u\) and \(u_i\) are positive and bounded below. Moving \(V_i+M_0\) to the other side, taking positive parts, and summing squares therefore yields \[ \sum_i t_i^2\leq \frac{C\epsilon^2}{r}\bigl(\mathbb EV_i^2+M_0^2\bigr) +C\sum_i(ND_i^0+t_i)_+^2 . \tag{65}\] The prescription error relation in Proposition 8 is \(ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i\), with \(\sum_i|\mathrm{err}_i|^2\) arbitrarily small. Moreover, the longitudinal component of \(\mathrm{II}_i\) is \(-X_i/N\), so \[ND_i+t_i=-(X_i-t_i) +N\bigl(\mathop{\mathrm{tr}}\mathrm{II}_i^{\mathrm{rem}}-U_i(\phi_{i-1})\bigr).\] The terms \((X_i-t_i)^2\), \(N^2|\mathrm{II}_i^{\mathrm{rem}}|^2\), and \(\beta_i^2f_i^2\) occur in \(\mathcal X\). For the remaining term, choose \(m_b<1/4\). Then \(L_i\leq C\) and \(|d\log L_i|_{h_i}\leq C A_i|\mathrm{II}_i|\). Because the metrics increase during the sweep and \(\phi_{i-1}=-\sum_{j<i}\log L_j\), Cauchy–Schwarz gives \[\begin{split} \sum_iN^2U_i(\phi_{i-1})^2 &\leq C m^2\sum_jN^2|\mathrm{II}_j|^2\\ &\leq C' m^2\left(\mathcal X+\sum_jt_j^2\right). \end{split}\] For example, the first line follows by bounding \(|d\phi_{i-1}|_{h_{i-1}}\) by \(C\sum_{j<i}A_j|\mathrm{II}_j|\), squaring with \(\sum_jA_j^2=m\), and then summing the factor \(A_i^2\). Choosing \(m_b\) small absorbs the last sum of \(t_j^2\) in (65). Since \[m^2=\left(\sum_i f_i t_i\right)^2\leq r\sum_i t_i^2, \qquad \epsilon^2(1-r)^2\leq C\bigl(\epsilon^2M_0^2+m^2\bigr),\] fractions of the \(M_0^2,\mathbb EV_i^2,\mathcal X\) terms in \(\epsilon\Pi_p\) control both required squares. The arbitrarily small errors in the prescription contribute \(r\delta'_0\). The term \(-4\theta^2m\) in (58) is \(-4l^2m\). This proves (64).

We now compare the right side of (64) with the required budget. When \(m\geq m_b\), the first two terms on the right of (63) are at most \(l\epsilon(m+4r)\). The squares \(m^2+\epsilon^2(1-r)^2\) dominate this expression with a fixed fraction to spare after \(l\) is chosen small and then \(\epsilon_b\) is decreased if needed. One may check this separately for \(r\leq2\), where \(m\geq m_b\) controls the constant part, and for \(r>2\), where Young’s inequality controls \(\epsilon r\) by \(\epsilon^2(r-1)^2\) and a sufficiently small fixed part of \(m^2\). The term \(C_n\epsilon l^2m\) is covered at the same time.

When \(m<m_b\), the identity \(L_i=(1-A_i^2)^{-1/2}\) gives \(W_p\leq C m\). For \(r\notin(1/2,2)\), Young’s inequality applied to \(l\epsilon m(1+r)\) again gives the required domination by the two squares, with \(l\) small. For \(1/2<r<2\), the positive terms on the right of (63), together with the term \(C_n\epsilon l^2m\) from the coercivity estimate, are at most \(C_nl\epsilon m\). Here \[C_nl\epsilon m\leq \frac{c_n}{2}m^2+C'_n l^2\epsilon^2.\] On this region \(r e^{-2W_p/(n-1)}\) has a dimensional positive lower bound. We may therefore choose \(C_b\) dimensionally to cover the last term. These comparisons can be made strict by increasing \(C_b\) and decreasing \(l,\epsilon_b\). They leave a positive multiple of \(l^2\epsilon^2r\) in the middle region, and positive square margins outside it; dividing the latter margins by \(r\) gives a positive lower bound for fixed \(l,\epsilon\). We can consequently choose \(\delta'_0\) small enough for the sweep. This proves (62).

The input map has a finite uniform energy bound in \(g\). Thus for some positive, possibly input-dependent \(\epsilon_{\mathrm{in}}\) small enough, \(\mathop{\mathrm{Scal}}_g=-1\geq-2+\epsilon_{\mathrm{in}}e_g(F^{\mathrm{in}})\). Choose a dimensional \(\epsilon_*>0\) below \(\epsilon_b\), small enough that \(2+2C_b\epsilon_*<4\), and decrease \(\epsilon_{\mathrm{in}}\) further so that \(0<\epsilon_{\mathrm{in}}<\epsilon_*\). Iterate (62) until the coefficient first reaches \(\epsilon_*\). The output coefficient is at most \((1+l)\epsilon_*\leq2\epsilon_*\), and \[\sum_{\text{bootstrap sweeps}}l\epsilon =\epsilon_0-\epsilon_{\mathrm{in}}\leq2\epsilon_*.\] This bounds the total scalar cost by \(2C_b\epsilon_*\), independently of the number of sweeps. By (61), every reset contracts the volume density. The final density \(\rho=dV_{h_*}\) therefore satisfies (59). The sweep construction and finitely many smooth conformal changes preserve all the stated input-uniform properties. ◻

Proof. The output energy bound in (57) shows that, in the units of \(Q\), it suffices to prove \[ Q\geq lm+2lrW_p-nrl\log(1+l)-C_nr l^3 +\frac{rN^2}{\epsilon}P_p . \tag{68}\] The term \(lm\) pays for the additional \(l\epsilon e_{h_+}(Y)\). We choose \(\delta_0\leq\epsilon l^3\) in (58). It remains to compare the other terms there with the budget in (68).

We use a trace identity for the amount of the new graph energy that remains in the final metric. Set \[m_f:=\sum_i A_i^2(1-C_i)=N^2\sum_i|df_i|_{h_+}^2.\] As matrices, \(\sum_iN^2df_i\otimes df_i=h_+-h_0\). Hence \[ m_f=n-\mathop{\mathrm{tr}}(h_+^{-1}h_0) \leq n(1-e^{-2W_p/n}). \tag{69}\] The inequality is arithmetic–geometric mean applied to the positive eigenvalues of \(h_+^{-1}h_0\), whose product is \(e^{-2W_p}\). We also have \[ L_i=(1-A_i^2)^{-1/2}. \tag{70}\]

First choose a small dimensional \(m_0>0\), in particular \(m_0\ll\epsilon_*\) and \(m_0<1/4\). We claim that \[ T_0+\epsilon M_0^2\geq c_n(m+rW_p) \quad\text{if }m\geq m_0 \text{ or if }m<m_0,\ |1-r|\geq\tfrac14. \tag{71}\] For \(m\geq2n\), the \(C_i\) terms of \(T_0\) give \(T_0\geq m-m_f\geq m-n\geq m/2\). For \(m_0\leq m\leq2n\), first suppose that every \(A_i^2\) is at most a small number \(\eta=\eta(n,m_0)>0\). Then \[2W_p=\sum_i-\log(1-A_i^2)\leq(1+C\eta)m.\] By (69), \[m-m_f\geq m-n\bigl(1-e^{-(1+C\eta)m/n}\bigr)\geq c_n m.\] For the last inequality, the same expression with \(\eta=0\) divided by \(m\) has a positive minimum on \([m_0,2n]\), and then \(\eta\) is chosen small. If instead some \(A_i^2>\eta\), the self term of that index in \(T_0\) is at least \(c_n(r+u_i)\). It gives a dimensional positive bound if \(r\) is bounded below. If \(r\) is small, \(M_0=a(1-r)-\bar u/2\) is bounded away from zero unless \(\bar u\) lies in a fixed compact subinterval of \((0,\infty)\). In the former case \(\epsilon M_0^2\geq c_n\epsilon_*\). In the latter, \(\epsilon=m/\bar u\leq C_n\) and \(u_i=A_i^2/(\epsilon\nu_i)\geq c_n\), since \(\nu_i\leq1\). Each bound controls \(m\leq2n\). This proves the \(m\) part for \(m\geq m_0\).

For the volume part, indices with \(A_i\geq1/4\) satisfy \[A_i^2(L_i-1)2ar\geq c\,r(L_i-1)\geq c\,r\log L_i.\] For all other indices, \(\log L_i\leq C A_i^2\), so their contribution to \(rW_p\) is at most \(Crm\). If \(r\leq2\), the bound for \(m\) already covers this. If \(r>2\), then \(|M_0|\geq c(r+\bar u)\), and \(\epsilon M_0^2\geq c\epsilon r\bar u=c\,rm\). This proves (71) when \(m\geq m_0\). When \(m<m_0\), we have \(\bar u\leq m_0/\epsilon_*\) small and \[W_p=\frac m2+O\left(\sum_iA_i^4\right)\leq C m.\] If \(|1-r|\geq1/4\), choosing \(m_0\) still smaller gives \(|M_0|\geq c|1-r|\). The quantity \(\epsilon(1-r)^2\) controls \(m+rm\): for bounded \(r\) use \(\epsilon\geq\epsilon_*\) and \(m\leq m_0\), and for \(r\geq2\) use \((1-r)^2\geq r^2/4\). This proves the claim in the remaining case.

On the region of (71), the positive reserve \(\Pi_p\) consequently controls \(m+rW_p\). The positive terms \(lm+2lrW_p\) in (68) use at most \(C_nl\Pi_p\), and \(4\theta^2m\) uses at most \(C_nD_*^2l^2\Pi_p\). For \(l\) sufficiently small after \(D_*\), a fixed fraction of \(\Pi_p\) remains. Choose \(\gamma_n\) below that fraction. The error term \(r\delta_0/\epsilon\) is covered by \(C_nr l^3\). This proves (68) on the region where the surplus is retained.

It remains to prove the budget when \[ m<m_0,\qquad |1-r|<\tfrac14. \tag{72}\] Here \(T_0\geq c_nm^2\). Indeed, expanding the exponential in (69) and using \(W_p=m/2+O(\sum_iA_i^4)\) gives \[\sum_i A_i^2 C_i=m-m_f \geq c_n m^2-C_n\sum_iA_i^4\] after decreasing \(m_0\). The self terms \(\sum_i A_i^2(L_i-1)2ar\) are at least \(c_n\sum_iA_i^4\), by \(r>3/4\) and \(L_i-1\geq A_i^2/2\). Adding a sufficiently small fraction of the first estimate to the second proves \(T_0\geq c_nm^2\). Since \(W_p\leq Cm\) on this region, this controls the budget whenever \(m\geq Kl\), for a fixed sufficiently large dimensional \(K\). In making this conclusion, first fix \(K\); after \(D_*\) is chosen below, take \(l\) small enough that \(4D_*^2l^2m\) is also a small fraction of \(m^2\).

We are left with the critical regime \(m\leq Kl\), still under (72). Here \(W_p=m/2+O(m^2)\) and \(3/4<r<5/4\), so the budget expands as \[lm+2lrW_p-nrl\log(1+l) =2lm-nl^2+O_n(l^3)+O_n(l^2)|r-1|.\] The target quadratic term is covered exactly by \[\frac{m^2}{n}-2lm+nl^2 =\left(\frac m{\sqrt n}-\sqrt n\,l\right)^2\geq0.\] We must therefore recover \(m^2/n\) from \(T_0\), while paying the displayed \(r-1\) error and the errors in that recovery. This is where the coefficient one on \(T_0\) in \(\Pi_p\) is needed.

In particular \(e^{W_p}=1+O(m)\). Metric addition gives \(h_0\leq h_j\leq h_+\) as quadratic forms. Every eigenvalue of \(h_0^{-1}h_+\) is at least one, and their product is \(e^{2W_p}\). Each eigenvalue is therefore at most that product, which proves the comparison \[h_0\leq h_j\leq h_+\leq e^{2W_p}h_0=(1+O(m))h_0.\] Normalize the directions \(e_i\) in the \(h_0\) inner product and write the resulting unit vectors as \(\bar e_i\). Put \(\xi_{ij}=(\bar e_i\cdot\bar e_j)^2\). Before approximating the metric losses, their exact earlier- and later-color telescopes are \[B_i=\sum_{j<i}(Ndf_j(e_i))^2,\qquad C_i=\frac1{A_i^2}\sum_{j>i}A_j^2(Ndf_i(e_j))^2 \quad(A_i>0).\] The first identity follows from \(h_{i-1}=h_0+N^2\sum_{j<i}df_j\otimes df_j\) and \(|e_i|_{h_{i-1}}=1\). The second follows by telescoping the rank-one inverse formula \[h_j^{-1}=h_{j-1}^{-1}-A_j^2 e_j e_j^{\mathsf T}\] from \(h_i\) to \(h_+\), and using \(N^2|df_i|_{h_i}^2=A_i^2\). Now \(Ndf_j=A_jL_j e_j^\flat\), with the dual taken in \(h_{j-1}\). The common metric comparison and \(L_j=1+O(m)\) show that each summand in the exact formulas differs from its \(h_0\) expression by \(O(m)A_j^2\). Their sums therefore give, for \(A_i\ne0\), \[ B_i=\sum_{j<i}A_j^2\xi_{ij}+O(m^2),\qquad C_i=\sum_{j>i}A_j^2\xi_{ij}+O(m^2). \tag{73}\]

Inserting (73) into \(T_0\) yields \[ T_0\geq \mathop{\mathrm{tr}}\left[\left(\sum_i A_i^2\bar e_i\bar e_i^{\mathsf T}\right)^2\right] -C_n\left(m^3+m^2|r-1| +m\sum_iA_i^2(1-\sigma_i)_+\right). \tag{74}\] To see the leading coefficient, the \(B_i\) and \(C_i\) terms each give one copy of \(\sum_{i<j}A_i^2A_j^2\xi_{ij}\). In the \(B_i\) terms use \(4a^2\geq1\), \(B_i\geq0\); replacing \(\sigma_i\) by one costs at most \(Cm\sum_iA_i^2(1-\sigma_i)_+\). The self terms use \(L_i-1\geq A_i^2/2\) and \(2a\geq1\). They give \(\sum_iA_i^4\), up to the displayed \(r-1\) and \(\sigma_i-1\) costs. The errors in (73) sum to \(O(m^3)\). This proves (74). The trace in that estimate is at least \(m^2/n\), since the matrix is positive semidefinite with trace \(m\).

The last error in (74) is paid for by the centered squares. From \(a(\sigma_i-1)=V_i-\lambda(u_i-\bar u)\), Cauchy–Schwarz and Young’s inequality give, for every fixed \(c'>0\), \[ \begin{split} m\sum_iA_i^2(1-\sigma_i)_+ &\leq C m\epsilon\,\mathbb E\bigl[u_i(|V_i|+|u_i-\bar u|)\bigr]\\ &\leq c'\epsilon\bigl(\mathbb EV_i^2+\theta\mathop{\mathrm{Var}}u\bigr) +C_{c'}\frac{m^2}{\theta}\epsilon(\bar u^2+\mathop{\mathrm{Var}}u). \end{split} \tag{75}\] Choose \(c'\) to use a small fraction of the squares in \(\Pi_p\). The coefficient of \(\epsilon\mathop{\mathrm{Var}}u\) in the last term, relative to \(\theta\epsilon\mathop{\mathrm{Var}}u\), is at most \(C_{c'}K^2/D_*^2\), because \(m\leq Kl\) and \(\theta=D_*l\). Choose \(D_*\) sufficiently large after \(K\) so that this term is absorbed as well. The remaining term is \[C_{c'}\frac{m^2}{\theta}\epsilon \bar u^2 =C_{c'}\frac{m^4}{\theta\epsilon}=O_n(l^3).\] Here \(\epsilon\geq\epsilon_*\), and \(K,D_*,\epsilon_*\) are already fixed dimensionally. Also \[|r-1|\leq C(|M_0|+\bar u).\] Consequently a term \(O_n(l^2)|r-1|\) is bounded by a small fraction of \(\epsilon M_0^2\) plus \(O_n(l^3)\): use Young’s inequality on \(l^2|M_0|\), and use \(\bar u\leq Kl/\epsilon_*\) on the other part. The errors \(m^3\), \(m^2|r-1|\), and \(4D_*^2l^2m\) in the present region are therefore covered with a remaining error \(O_n(l^3)\).

The just established mean-square estimate also covers the \(O_n(l^2)|r-1|\) term in the critical budget. The recovered \(m^2/n\) then covers \(2lm-nl^2\) by the displayed square. Since \(r\) is bounded above and below, the remaining \(O_n(l^3)\) terms are of the form \(C_nr l^3\). This proves (68) in (72).

The choices just made have a definite order: first \(m_0\) after \(\epsilon_*\), then \(K\), then \(D_*\), and finally \(l_0>0\) small enough for all the comparisons and for \(D_*l_0\leq1/16\). The errors in each sweep are chosen afterward. Thus every constant in (66) is dimensional. ◻

Proof. If \(\bar u=m/\epsilon\) is larger than a fixed dimensional number, then \(m\geq m_0\), so the surplus retains the mean square and boost. Since \(|M_0|\geq c(r+\bar u)\) in this range and \(l\leq1\), \[P_p\geq c_n\epsilon\, \frac{(r+\bar u)^2+c_*r^6\mathbf 1_{\{r\geq C_*\}}}{r} \geq c'_n\epsilon \bar u^{5/3}.\] For the last inequality, if \(r\leq \bar u^{1/3}\) use \(\bar u^2/r\); if \(r\geq \bar u^{1/3}\) use \(r^5\), taking the threshold for \(\bar u\) large enough that \(\bar u^{1/3}\geq C_*\). It follows in all cases that \[ m\leq C_n\epsilon(1+P_p/\epsilon)^{3/5}. \tag{78}\] If any \(A_i>1/2\), then \(m>1/4>m_0\), and the self terms in \(T_0\) give \[P_p\geq c_n\sum_{A_i>1/2}(L_i-1).\] Indeed their contribution to \(T_0\) is at least \(c r\sum_{A_i>1/2}(L_i-1)\), and \(\epsilon/(rN^2)=1/(lr)\geq1/r\). For \(A_i\leq1/2\), \(\log L_i\leq C A_i^2\). There are at most \(4m\) indices with \(A_i>1/2\). Concavity of \(\log(1+s)\) on those indices, together with the last estimate, therefore gives \[ W_p\leq C_n m\bigl(1+\log(1+P_p/m)\bigr) \tag{79}\] when \(m>0\); when \(m=0\), \(W_p=0\). To justify the bound when the number of those indices is smaller than \(4m\), use that \(s\log(1+b/s)\) is increasing in \(s>0\). Put \(x=m/\epsilon\), \(y=P_p/\epsilon\). The elementary bound \(1+\log(1+t)\leq C(1+t)^{1/4}\) for \(t\geq0\), applied with \(t=y/x\), makes the right side of (79), divided by \(\epsilon\), at most \[C_n x^{3/4}(x+y)^{1/4} \leq C_n(1+y)^{7/10} \leq C_n(1+y)^{3/4},\] where (78) was used in the middle inequality. This proves (76).

Now suppose \(r\leq1/4\). The surplus retains all positive terms because \(|1-r|\geq3/4\). Outside a fixed compact subinterval of \(\bar u\in(0,\infty)\), the expression \(M_0=a(1-r)-\bar u/2\) satisfies \(M_0^2\geq c(1+\bar u^2)\). On that compact interval, for \(\epsilon\) sufficiently large, \[T_0\geq m-m_f\geq\epsilon \bar u-n \geq c_n\epsilon(1+\bar u^2)\] by (69). Multiplying the appropriate retained term by \(\gamma_n/(lr)\) proves the first inequality in (77).

For the second, the retained derivative terms imply \[ \mathcal X\leq C_n N^2P_p,\qquad \frac{\epsilon^2}{rN^2} \bigl(\mathbb EV_i^2+\theta\mathop{\mathrm{Var}}u\bigr)\leq C_nP_p. \tag{80}\] Also \[\sum_i t_i^2=\frac{\epsilon^2}{r}(\bar u^2+\mathop{\mathrm{Var}}u).\] The first inequality of (77) controls the \(\bar u^2\) part of this sum divided by \(N^2\), and the variance term in (80) controls the other part with a factor \(1/\theta\). Thus \(\sum_i t_i^2/N^2\leq C_nP_p/\theta\). Since \(\mathcal X\) contains \((X_i-t_i)^2+N^2|\mathrm{II}_i^{\mathrm{rem}}|^2+N^2|d\log L_i|^2\), this proves the claimed bounds for \(\mathrm{II}_i\) and \(d\log L_i\).

For \(D_i\), use \[ND_i^0=\frac{\epsilon f_i}{r} \left(a(\sigma_i-r)+\frac{u_i}{2}\right),\qquad \mathbb E\sigma_i^2\leq C\bigl(1+\mathbb EV_i^2+\mathop{\mathrm{Var}}u\bigr).\] Here the second inequality follows by solving the definition of \(V_i\) for \(\sigma_i-1\). Since \(r\leq1/4\), \[\sum_i(D_i^0)^2\leq \frac{C\epsilon^2}{rN^2} \bigl(1+\bar u^2+\mathbb EV_i^2+\mathop{\mathrm{Var}}u\bigr) \leq C_nP_p/\theta.\] The relation \(ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i\) finishes the estimate: \(\sum_i\beta_i^2f_i^2\leq\mathcal X\), and we choose \(\sum_i|\mathrm{err}_i|^2\leq1\). Its contribution \(1/N^2\) is bounded by \(C_nP_p\) using the first inequality of (77), \(\epsilon\geq1\), and \(r\leq1/4\). ◻

Proof. The accumulation assertion follows by adding (66). The energy coefficients telescope multiplicatively, and \(\sum_p\log(1+l(\epsilon_p))=\log(E/\epsilon_{\mathrm{in}})\). The density identity in (57) gives the sum of the \(W_p\).

Put \(t_p=\log(1+\epsilon_p)\). Since \(\epsilon_p\geq\epsilon_*>0\) and \(l_*\) is fixed small, \[t_{p+1}-t_p =\log\left(1+l(\epsilon_p)\frac{\epsilon_p}{1+\epsilon_p}\right) \asymp_n l(\epsilon_p) =\frac{l_*}{(1+t_p)^2}.\] Comparison with the integrals of \((1+t)^2\) and \((1+t)^{-2}\) respectively proves the number bound and \[\sum_p l(\epsilon_p)^2 \leq C_n l_*\int_{t_0}^{\infty}\frac{dt}{(1+t)^2}\leq C_n.\] The comparison is valid on each interval \([t_p,t_{p+1}]\), since its endpoints differ by a uniformly bounded small number.

Let \(k\) be the number of sweeps in the group under consideration. By Lemma 12, since \(\epsilon_p\leq E\) and \(P_p\leq P\), \[\sum_pW_p\leq C_n k E^{1/4}(E+P)^{3/4}.\] For all sufficiently large \(E\), \(C_n k\leq \tfrac12 E^{0.01}\) by (82). This proves (83) for zero initial volume logarithm. In the other case \(P\geq P_0\) and \(E\geq E_0\), so the initial term is at most \(E_0^{0.32}(E+P)^{0.75}\). The stated condition on \(E_0\) pays for this with the other half of \(E^{0.26}(E+P)^{0.75}\). ◻

Proof. In this proof all derivatives, unless otherwise indicated, use \(h_+\). Write \(r=\sum_i f_i^2\) for the last sweep, set \(s=\sqrt r\), and put \[c_E=\frac1{\sqrt E(1+K_0)},\qquad Z=\frac{(f,c_EY^0)}{\sqrt{r+c_E^2}}.\] The two vectors in the numerator use disjoint copies of the labels. The map is a smooth root probability map because its denominator is at least \(c_E>0\). Writing the copies as \((i,0)\) and \((i,1)\), we call \(i\) their common parent label at the start of the phase; the sweeps within the phase do not change the label set. The first copy is supported on the last bumps and the second on the starting map. This parent assignment will be used to compare support overlaps between phases. By (69), \[N^2\sum_i|df_i|_{h_+}^2=m_f\leq n,\qquad |ds|_{h_+}^2\leq n/N^2.\] Monotonicity of the phase metrics also gives \(e_{h_+}(Y^0)\leq K_0^2\).

Since \(Y\) and \(Y^0\) are unit vectors in orthogonal spaces, their derivatives are perpendicular to their respective vectors. Differentiating \(Z=(sY,c_EY^0)/\sqrt{s^2+c_E^2}\) thus gives the exact identity \[e(Z)=\frac r{r+c_E^2}e(Y)+\frac{c_E^2}{r+c_E^2}e(Y^0) +\frac{c_E^2}{(r+c_E^2)^2}|ds|^2.\] For \(r\geq1/4\), it follows that \[E(e(Z)-e(Y))\leq \frac{Ec_E^2K_0^2}{r+c_E^2} +\frac{Ec_E^2n}{N^2(r+c_E^2)^2} \leq 4+\frac{16n}{N^2}\leq C_n\] when \(E\) is large, since \(N^2=l\epsilon\) tends to infinity under (81). For \(r\leq1/4\), the norm of the derivative of normalization is at most \(1/\sqrt{r+c_E^2}\), so \[ e(Z)\leq\frac{n/N^2+c_E^2K_0^2}{r+c_E^2}. \tag{86}\] The first inequality of (77) now implies \[E e(Z)\leq \frac{C_n}{l(r+c_E^2)} \leq \frac{C_n}{\epsilon}P_p.\] Thus replacing \(Y\) by \(Z\) in (84) costs a dimensional constant on \(\{r\geq1/4\}\) and an arbitrarily small fraction of \(P_p\) on \(\{r\leq1/4\}\), once \(E\) is dimensionally large. In particular, no factor \(K_0\) occurs in this scalar cost.

Choose a smooth function \(u=u(r)\geq0\), supported in \(\{r\leq E^{-0.32}\}\), with \[ e^u\asymp \min\bigl(E^{0.02},\max(1,E^{-0.16}/\sqrt r)\bigr), \qquad \left|\frac{du}{dr}\right|\leq\frac C r. \tag{87}\] For example, smooth, with bounded-width transitions on the logarithmic scale, the function obtained by truncating \(-\tfrac12\log r-0.16\log E\) to the interval \([0,0.02\log E]\). The smoothing may vanish whenever the untruncated function is nonpositive, and changes its value by a bounded amount; this gives both the support assertion and the comparison in (87). Define \[\widehat h=e^{2u}h_+,\qquad \widehat\phi=\phi_+-(n-1)u,\qquad \widehat W=W+nu.\] In particular \(dV_{\widehat h}=e^{nu}dV_{h_+}\), so this is the claimed volume logarithm. Differentiating \(r\) gives \[|dr|\leq2\sqrt r\left(\sum_i|df_i|^2\right)^{1/2} \leq\frac{2\sqrt{nr}}N,\qquad |du|\leq\frac{C_n}{N\sqrt r}.\] The coefficient \(n-1\) in the new weight cancels the Laplacian of \(u\) in the weighted conformal formula. With \(b=\nabla_{h_+}\phi_+\), that formula is \[ \mathcal S(\widehat h,\widehat\phi) =e^{-2u}\left[\mathcal S(h_+,\phi_+)+2b(u)-(n-1)|du|^2\right]. \tag{88}\]

With the fixed density \(\rho\), a graph sweep changes \(W\) by \(+W_p\) and \(\phi\) by \(-W_p\), so it preserves \(\phi+W\). The present exit instead satisfies \[\widehat\phi+\widehat W=\phi_++W+u.\] Thus after an exit \(W\) need not equal \(-\phi\), and its volume bound supplies no bound for the gradient of the weight. We need a bound on \(b(u)\) which uses no bound on the size of \(b\). It is enough to estimate this on the support of \(u\), which lies in \(\{r\leq1/4\}\) for large \(E\). At such a point view all the graphs of the last sweep in the common Euclidean extension of \((T_x\widetilde M,h_0)\) by their graph directions, one for each active color at that point. Write \(\mathbf n_i\) for the successive unit normals. They are orthonormal: a later normal lies in the preceding graph plus its new vertical direction and is perpendicular to every earlier normal. In this space, each gradient is embedded through its graph and padded by zero in the unused later directions. Writing \(b_i\) for the embedded gradient of \(\phi_i\), orthogonal projection to the new graph gives the exact recurrence \[b_i=b_{i-1}-\langle b_{i-1},\mathbf n_i\rangle\mathbf n_i -\nabla_{h_i}\log L_i,\] where the last gradient is embedded in the same space. The normal coefficient is \(\langle b_{i-1},\mathbf n_i\rangle=-U_i(\phi_{i-1})\), and its magnitude satisfies \[|U_i(\phi_{i-1})|\leq |D_i|+\sqrt n\,|\mathrm{II}_i|,\] because \(D_i=\mathop{\mathrm{tr}}\mathrm{II}_i-U_i(\phi_{i-1})\). By orthonormality and (77), the norm of the sum of these normal corrections is at most \(C_n\sqrt{P_p/\theta}\).

For the subtracted tangent gradients, split the indices according as \(A_i\leq1/2\) or \(A_i>1/2\). In the first set \(|d\log L_i|\leq C A_i|\mathrm{II}_i|\); in the second there are at most \(4m\) indices. Cauchy–Schwarz and (77) consequently bound the sum of their norms by \(C_n\sqrt{mP_p/\theta}\). The final tangent embedding sends a vector \(v\) to \((v,(Ndf_i(v))_i)\). Summing the recurrence, the initial gradient has no vertical component in this common extension, while the vertical component of the final gradient is \((Ndf_i(b))_i\). The normal and tangent bounds therefore prove \[\sum_i(Ndf_i(b))^2\leq C_n(1+m)P_p/\theta.\] It follows that \[|b(u)|\leq\frac C r|b(r)| \leq C_n\left(\frac{(1+m)P_p}{\theta N^2r}\right)^{1/2} \leq \frac{C_nP_p}{\sqrt{\theta E}}.\] For the last inequality use \(E\asymp\epsilon\), \(N^2=l\epsilon\), and \[P_p\geq\frac{c_n}{lr} \left(\epsilon+\frac{m^2}{\epsilon}\right) \geq \frac{c'_n(1+m)}{lr}\] for \(\epsilon\geq1\). The same first surplus bound gives \[ |du|^2\leq\frac{C_n}{N^2r}\leq\frac{C_nP_p}{E^2}. \tag{89}\] All these estimates are only needed where \(u\ne0\), a subset of \(\{r\leq1/4\}\) for large \(E\). Since \(\theta E\asymp_n E/(1+\log E)^2\) tends to infinity, the two derivative terms in (88) cost an arbitrarily small fraction of \(P_p\).

There is enough surplus on this region also for the change in the entropy term. From \(r\leq E^{-0.32}\) and (77), \[P\geq P_p\geq c_n E^{1.32}.\] Using (84) and \(e^{2u}\leq C E^{0.04}\), we obtain there \[ \frac{e^{2u}W}{P}\leq C_nE^{0.04+0.26}P^{-0.25}\leq C_nE^{-0.03}, \qquad \frac{e^{2u}nu}{P}\leq C_nE^{-1.28}\log E. \tag{90}\] In the first bound we used \(E+P\leq C_nP\), valid for large \(E\) on this region. Both quantities tend to zero with dimensional constants.

We now assemble these estimates. After replacing \(Y\) by \(Z\), write the retained surplus as \(\alpha_n P\), where \(\alpha_n>0\) is fixed dimensionally. Put \(B_u=2|b(u)|+(n-1)|du|^2\). Substituting in (88) and moving the new entropy term to the left gives \[\begin{align*} &\mathcal S(\widehat h,\widehat\phi) -E e_{\widehat h}(Z)-2(W+nu)+n\log E\\ &\quad\geq -(C+C_n) +e^{-2u}\bigl[\alpha_n P-B_u -2(e^{2u}-1)W-2nu e^{2u}\bigr] +(1-e^{-2u})n\log E . \end{align*}\] We used \(e^{-2u}(C+C_n)\leq C+C_n\). The last term is nonnegative for \(E\geq1\). Equations (89) and (90) leave at least a fixed fraction of \(P\) in the bracket on the support of \(u\); off that support the volume and derivative costs vanish. We may thus set \[P'=\alpha'_n e^{-2u}P\] for a fixed \(\alpha'_n>0\), proving the scalar inequality in (85). The only scalar constant charged was the dimensional replacement cost.

The volume estimate at exit follows as well. Since \(P\leq C_ne^{2u}P'\leq C_nE^{0.04}P'\) and \(nu\leq C_n\log E\), \[\widehat W\leq C_n E^{0.26+0.04(0.75)}(E+P')^{0.75}+C_n\log E \leq E^{0.32}(E+P')^{0.75}\] for all sufficiently large dimensional \(E\).

Finally, (86) and the conformal scaling of energy give \[e_{\widehat h}(Z)^{1/2} \leq e^{-u}\frac{\sqrt n/N+c_EK_0}{\sqrt{r+c_E^2}}.\] The choice (87) implies \[\frac{e^{-u}}{\sqrt{r+c_E^2}} \leq C\left(\frac{E^{-0.02}}{c_E}+E^{0.16}\right).\] Using \(\sqrt E/N\leq C_n(1+\log E)\), \(c_EK_0\leq E^{-1/2}\), and \(N\leq\sqrt E\), we conclude that \[ e_{\widehat h}(Z)^{1/2} \leq C_n\left( \frac{(1+K_0)(1+\log E)}{E^{0.02}}+\frac{E^{0.16}}{N}\right). \tag{91}\] The first summand is \(o(E^{-0.01})\) for fixed \(K_0\); the second is \(O_n((1+\log E)E^{-0.34})\). Taking \(E\) large enough in terms of \(n,K_0\) proves the final inequality in (85). The functions \(u\) and the normalizing denominator are smooth with finite bounds in each fixed construction. The supports of \(Z\) are those of the last bumps and of \(Y^0\). Thus the asserted uniformity and compact support properties persist. ◻

Proof of Theorem 9. Fix parameters \(H,A,B\) for the prescribed elementary bound and set \(J=H+2\). This number of phases is fixed before any phase target is chosen. The first target will be chosen after the bootstrap data and their uniform constants have been fixed; the estimates below hold for every sufficiently large choice of that target.

Apply Lemma 10. At its output \(W=\log(dV_{h_*}/\rho)=0\), \(\phi=0\), and \(\epsilon_0\in[\epsilon_*,2\epsilon_*]\). Increasing a dimensional constant once, (59) has the form \[\mathcal S(h_*,0)\geq-C_n+2W-n\log\epsilon_0 +\epsilon_0 e_{h_*}(F_*),\] because \(\epsilon_0\) ranges in a fixed dimensional interval. We use the schedule (81) from now on. The bound (82) applies to the entire increasing sequence of energy coefficients, even when it is partitioned into phases.

Choose a large first target and run the first phase until the parameter first reaches it; call the actual output parameter \(E_1\). The overshoot is at most a factor \(1+l_*\). In every sweep of this phase use \(h_*\) as the reference metric. The possible support enlargement per sweep, including its separation buffers, has a dimensional radius in this reference metric by Proposition 8. The total radius is therefore at most \(C_n(1+\log^3(1+E_1))\).

We record a bound for a single fixed graph sufficient for every coloring in this phase and for all cooccurrences at its exit. All data before the choice of the first target, including \(h_*\) and its uniform comparison with \(g\), have already been fixed. The lifted metric \(h_*\) is at least a positive input-dependent multiple of \(g\). For a fixed finite generating set of \(\Gamma\), cocompactness implies that word length is bounded above by an affine input-dependent function of lifted \(g\)-distance between translates of fixed compact fundamental sets. For instance, subdividing a curve into bounded length pieces and following the finitely many translates meeting successive covering sets gives this bound. The number of group elements of word length at most \(L\) is at most exponential in \(L\). It follows that the graph joining every pair of labels whose possible enlarged supports in the first phase meet, or come within a required buffer, has degree at most \[ \Delta_1+1\leq \exp\bigl(C_{\mathrm{inst}}(1+\log E_1)^3\bigr). \tag{92}\] Here \(C_{\mathrm{inst}}\) is finite and fixed before the first target is chosen. It includes the initial number of label orbits and the compact support bounds. The two copies of labels used at exit and their possible cooccurrences change this bound only by a fixed factor, which is absorbed in \(C_{\mathrm{inst}}\).

Take \(E_1\) sufficiently large for Lemma 13 and Lemma 14; in the latter the initial \(K_0\) is the finite input-dependent gradient bound of \(F_*\). We now have the scalar inequality, the volume bound with exponent \(0.32\), and the absolute gradient bound at \(E_1\). Denote the exit metric, map, and fixed cooccurrence graph by \(h_1,F_1,\mathcal G_1\).

We next isolate how a small gradient controls the graph in a later phase. Suppose a phase starts with metric \(h_j\), root map \(F_j\) with \(e_{h_j}(F_j)^{1/2}\leq K\), and a fixed graph \(\mathcal G_j\) containing all its possible cooccurrences. Let \(R\) bound the following pairwise distance: whenever two possible enlarged supports meet or come close enough to require separation by a buffer, there are points \(x,y\) in the old nonzero supports of their respective parent labels with \(\mathop{\mathrm{dist}}_{h_j}(x,y)\leq R\). If a closure is involved, allow an arbitrarily small additional distance. Suppose \[KR<\sqrt2.\] Choosing the additional distance small enough, the gradient bound gives \[\|F_j(x)-F_j(y)\|_{\ell^2} \leq K\mathop{\mathrm{dist}}_{h_j}(x,y)<\sqrt2.\] Two nonnegative unit vectors with disjoint supports have distance exactly \(\sqrt2\). Thus \(F_j(x)\) and \(F_j(y)\) share a nonzero label, say \(k\). The first parent and \(k\) cooccur at \(x\), and the second parent and \(k\) cooccur at \(y\). The two parents are therefore at graph distance at most two in \(\mathcal G_j\). This argument applies separately at each transverse parameter; because \(\mathcal G_j\) already contains all possible cooccurrences, the conclusion holds in that one fixed graph.

After an exit at \(E_j\), we choose a target that both meets this overlap condition and grows by one exponential level: \[ T_{j+1}=\exp(E_j^{1/1000}), \tag{93}\] and call the first parameter reaching it \(E_{j+1}\). Thus \[T_{j+1}\leq E_{j+1}\leq(1+l_*)T_{j+1}.\] The choice \(1/1000\) leaves the strict inequality \(3/1000<1/100\): the cubic logarithmic cost of support enlargement will be smaller than the gain from the preceding gradient bound. For large \(E_j\) this target satisfies \(E_j^{0.32}\leq\tfrac12 E_{j+1}^{0.26}\). The outgoing bound with exponent \(0.32\) from phase \(j\) and Lemma 13 therefore give the bound with exponent \(0.26\) before the next exit. The next initial gradient bound is \(K_0\leq E_j^{-1/100}\leq1\), so all further thresholds for Lemma 14 are dimensional. Increasing the first target once ensures these conclusions in every one of a prescribed finite number of phases.

Use \(h_j\), the starting metric, as the reference metric in phase \(j+1\). Let \(R_{j+1}\) bound the pairwise old-support distance in the preceding argument, including both support enlargement radii and all separation buffers. The number of sweeps gives \[R_{j+1}\leq C_n(1+\log^3 E_{j+1}) \leq C_n E_j^{3/1000}.\] Consequently \[ R_{j+1}E_j^{-1/100}\leq C_n E_j^{-7/1000}\longrightarrow0. \tag{94}\] For large \(E_j\), the overlap condition therefore holds with \(K=E_j^{-1/100}\). For every relevant pair the preceding estimate becomes \[\|F_j(x)-F_j(y)\|_{\ell^2} \leq E_j^{-1/100}\mathop{\mathrm{dist}}_{h_j}(x,y)<\sqrt2.\] This uses only the starting metric of the phase, without a comparison constant between that metric and \(g\).

It follows that every separation graph needed in phase \(j+1\) is contained in the square \(\mathcal G_j^2\), after identifying labels with their parents at the start of the phase. At exit there are two copies of each parent. Joining two such copies whenever their parents are equal or at distance at most two gives a fixed graph \(\mathcal G_{j+1}\) containing all new cooccurrences. If \(\Delta_j\) is the degree of \(\mathcal G_j\), this construction gives \[ \Delta_{j+1}+1\leq2(\Delta_j+1)^2. \tag{95}\] It also doubles the number of label orbits. The preliminary coloring construction applies to these graphs before each sweep. All analytic support and chart bounds in the resulting finite construction remain finite for the input, even though they need not satisfy a useful quantitative bound. They do not enter (95).

After \(J\) phases, apply the coloring argument to \(\mathcal G_J^2\), whose maximum degree is at most \(\Delta_J^2\). Take \(q\) to be the maximum of its degree plus one, the number of label orbits, and \(n+1\). Equations (92) and (95) imply \[ \log q\leq C_{\mathrm{inst},J,n}(1+\log E_1)^3. \tag{96}\] For fixed \(J,n\) the coefficient here is fixed before \(E_1\) is chosen. The graph \(\mathcal G_J\) and this coloring have exactly the properties asserted in the theorem.

It remains to make the first target large enough for the already fixed \(J=H+2\). Let \(\mathcal B(q)\leq\exp^{[H]}(B(1+q)^A)\) be the prescribed elementary bound. Equation (96) implies \[\mathcal B(q)\leq \exp^{[H+1]}\bigl(C'_{\mathrm{inst},J,n,A,B} (1+\log E_1)^3\bigr)\] for large \(E_1\). On the other hand, the recurrence (93) gives, for each fixed \(J\geq2\), \[ E_J\geq\exp^{[J-1]}\bigl(c_J E_1^{1/1000}\bigr) \tag{97}\] for some \(c_J>0\) and all sufficiently large \(E_1\). One proves this by induction: raising a fixed-height exponential tower with argument \(c t\) to the power \(1/1000\) still dominates a tower of the same height with argument \(c't\), for some \(c'>0\) and all large \(t\); the next application of \(\exp\) adds a level. For height one this follows from \((e^{ct})^{1/1000}=e^{ct/1000}\); for higher heights, reducing the bottom argument by any fixed factor absorbs the fixed outer multiplicative factor. This proves the induction.

With the already chosen \(J=H+2\), the tower heights on the two sides are now the same, and \[c_J E_1^{1/1000}> C'_{\mathrm{inst},J,n,A,B}(1+\log E_1)^3\] for all sufficiently large \(E_1\). Thus \(E_J>\mathcal B(q)\). We may choose the first target still larger to have \(E_J>E_{\min}\), to satisfy the initial gradient threshold involving \(K_0\), and to meet all dimensional thresholds used above. These choices affect only the size of the final parameter and the finite uniform bounds of the output objects.

Finally we account for the scalar constant. The bootstrap charges at most \(2C_b\epsilon_*\), independently of its input-dependent number of sweeps. The initial logarithmic normalization has a dimensional cost because \(\epsilon_0\in[\epsilon_*,2\epsilon_*]\). All entropy sweep losses together are bounded by \(C_n\sum l^2\leq C_n\). Each exit charges at most \(C_n\), and their number \(J=H+2\) depends only on the prescribed elementary function and \(n\). The input gradient bound \(K_0\), the initial small \(\epsilon_{\mathrm{in}}\), the compact support radii, and all metric and elliptic bounds occur only in the choices of the first target, graphs, and arbitrarily small analytic errors. None contributes to the scalar constant. Discard the final nonnegative surplus \(P'\); the remaining inequality is (56) with \(C=C(n,\mathcal B)\).

The density \(\rho\) has been unchanged since the bootstrap, so \(\rho\leq dV_g\). The volume logarithm starts at zero and only increases by \(W_p\geq0\) and \(nu\geq0\). Finitely many sweeps and exits preserve the asserted smoothness, uniformity, compact supports, and finite number of free label orbits. The transverse spaces were enlarged only by the probability products used for the colorings, and the sweep constructions are Borel on these spaces by Proposition 8. This proves all the conclusions. ◻

Proof of Theorem 16. The case \(d=0\) is the signed-count inequality in the statement. Suppose the theorem is known in dimension \(p=d-1\), with the same \(q,B,\tau\) and the residual lengths. Its hypotheses hold for the child of Lemma 17, including the unchanged strict horizon condition. For almost every \(s\) the induction hypothesis and (105) give \[|\mathcal D_X(G)|(2\tau)^p \le \frac{e^{C_1p/q+C_*/q}}{\prod_{i=1}^dL_i} \int_{\Sigma_s\cap\{|G|_\infty<b_d\}}v_s\Theta\,dA_h.\] For \(p=0\) this is the same use of the unsigned-count bound. Take \(C_1\ge C_*\). Integrate this scalar inequality over \(-\tau<s<\tau\) and apply (104) with \(F=\mathbf1_{\{|G|_\infty<b_d\}}\Theta\). The factor \(2\tau\) on the left changes its power from \(p\) to \(d\), and the right is bounded by the integral in (103). This proves the induction.

There is no implicit measurable selection of children in this last step. For each typical \(s\) a child proves the displayed scalar inequality against the same parent integral. That parent integral is a measurable function of \(s\) by the lapse construction and is governed by (104); it is the only function being integrated. Its integral is finite because \(\Theta\) is bounded on the controlled \(b_d\) region, which has finite measured volume. The constants \(C_0\) and \(C_1\) are absolute: \(C_0\) came from the fixed pressure profile, and each descent costs at most \(e^{C_*/q}\) after the arbitrarily small slice errors were chosen. All geometry-dependent constants were used only to make those choices. ◻

Proof. Take one copy of \(B(z_y,10\delta)\subset\mathbb R^q\) for each witness \(y\in K\). At the same point \(z\) in two such balls, identify the two copies if their witnesses share a label in their nonzero supports. We check that these identifications give ball charts, rather than a branched space. For every point under consideration, some coordinate of \(z\) is at least \(q^{-1/2}-10\delta\). Every witness for that point has a nonzero label at this color. If two witnesses share any label, their labels at every common color agree: each of the two labels at that color is adjacent to the shared label, so distance-two separation of colors forces them to be the same. Conversely, agreement at the indicated large color is already a shared label. Thus, at \(z\), the proposed identification is precisely agreement on the label at that large color, which is an equivalence relation.

The same color remains positive in all witnesses at every sufficiently nearby point; indeed the margin \(q^{-1/2}-21\delta>0\) suffices after a further displacement of at most \(\delta\). In that neighborhood the equivalence classes are still distinguished by its label. Each ball therefore projects injectively as a local chart. Points with distinct Euclidean projections can be separated in Euclidean space, and points with the same projection and different labels at the large color have disjoint chart neighborhoods just described. This proves the Hausdorff property. Rational dense choices of centers in each spherical orthant give a countable subatlas: there are countably many labels and cliques. The descriptions by labels, clique coordinates, and color assignments are Borel, and are unchanged by simultaneous translation of labels and \(\omega\). They prove all the claimed family and equivariance properties. At points of \(K\) this also shows that its inclusion is unambiguous.

The lift of \(F\) is locally in one of these charts: its label at a chosen large color persists in a neighborhood. A nonnegative smooth root coordinate has zero differential wherever it is zero. All the nonzero coordinates at a point have different colors. It follows that the nonzero rows of \(dF\) are merely moved to different color positions by \(dz_F\), proving the asserted energy equality.

To construct this cutoff pullback, choose a nonnegative smooth function \(\chi\) which is zero for \(t\leq12\delta\), equals \(t\) for \(t\geq25\delta\), lies between zero and \(\max\{t,0\}\), and satisfies the usual rescaled derivative bounds \(|\chi^{(j)}|\leq C_j\delta^{1-j}\) for \(j\geq1\). In a ball put \[w(z)=\frac{(\chi(z_j))_{j=1}^q}{|(\chi(z_j))_{j=1}^q|}, \qquad p_j=w_j^2.\] The denominator is bounded below by an absolute positive constant. In fact \(z\) is within \(10\delta\) of a nonnegative unit vector, and the change in each coordinate caused by cutting it off is at most \(25\delta\); hence its total change is at most \(25\delta\sqrt q\). If \(\chi(z_j)\ne0\), every witness in this ball has a positive coordinate at color \(j\), because its distance from \(z\) is less than \(10\delta\). The label for every active coordinate is therefore unambiguous on chart overlaps. Evaluate \[\eta=n!\,c(p,dp,\ldots,dp)\] using those labels. The cocycle law makes this form closed, as in Proposition 4, and a straight probability homotopy from the old probabilities to these new ones preserves the measured pairing. Its local finiteness and smooth bounds are uniform for the instance, so the measured Stokes argument of that proposition applies. This gives the first identity in (144).

For the dimension-only norm bound, first regard the form on nonnegative unit root vectors \(w\). In the Euclidean coordinate directions \(i_1,\ldots,i_n\), its component is \[n!2^n\Bigl(\prod_{j=1}^n w_{i_j}\Bigr) \sum_a w_a^2 c(a,i_1,\ldots,i_n).\] Its absolute value is at most \(n!2^n\prod_j w_{i_j}\). The sum of the squares of these products over all increasing \(n\)-tuples is at most \((\sum_i w_i^2)^n=1\). Thus its exterior Hilbert norm is at most a constant depending only on \(n\). The map \(z\mapsto w(z)\) has absolutely bounded operator norm of its differential: its denominator is bounded below, normalization is an orthogonal projection divided by that denominator, and \(\chi'\) is absolutely bounded. Pullback multiplies the exterior Hilbert norm by at most the \(n\)-th power of this operator bound. This proves the second assertion in (144). Differentiating the same formulas a fixed number of times gives elementary bounds (polynomial bounds in \(q\) already suffice), since \(\delta^{-1}=1000\sqrt q\). ◻

Proof. The closure equations couple faces whose anchors can differ. We encode all local coefficients in one ordered finite vector and round within the joint kernel of its incidence equations, so every cell reads from the same rounded vector. We first establish that the vector has elementary size despite the infinitely many lifted cells.

All projected centers lie in a fixed bounded Euclidean ball. For fixed \(V,s\), their grid indices and axis sets therefore have at most \((C/r)^{Cq}\) choices. At a selected center choose, breaking ties by color order, its largest unrotated color coordinate. This coordinate is at least \(q^{-1/2}-4\delta\), and its label occurs in every witness for the center. It determines the lift uniquely, by the proof of Lemma 19; call it the anchor label.

The closed graph one-neighborhood of an anchor has at most \(q\) labels. Every center-test witness, meaning a witness at distance less than \(4\delta\) from the center, lies in this neighborhood because it contains the anchor. Every point of the full cell is within \(4\delta+r\sqrt n/2<5\delta\) of each such witness. Thus every active color, whose coordinate is greater than \(12\delta\), occurs in each center-test witness. For an arbitrary ball-chart witness of the center the corresponding bound is instead \(10\delta+r\sqrt n/2<11\delta\), which is still below the active cutoff and gives the same conclusion. Thus the coefficient \(u_C\) is determined by its projected grid index, its anchor, and the color assignment on this one-neighborhood. In particular, no labels farther out in the graph enter its calculation.

Choose and order representatives of the at most \(q\) anchor orbits, and order each of their finite one-neighborhoods. An anchor has a unique translate to its representative because the label action is free. After this translate, its neighborhood is a fixed finite set and the cocycle evaluations are unchanged by homogeneity. Fix one ordered list of slots consisting of an anchor representative, an injective color assignment on its neighborhood, an axis set and grid index, and a witness clique in that neighborhood containing the anchor. It is enough to use the fixed index box \(|k_j|\leq\lceil3/r\rceil+3\), which contains every possible selected center for all shifts. The slot is valid when the anchor color is the canonically chosen largest unrotated color of its center and that center is at distance less than \(4\delta\) from the spherical orthant of its witness clique. Give a valid slot the integral of the explicit form over its primal cell, using the labels specified by the slot, and give an invalid slot value zero. The preceding active-color observation makes this integral unambiguous. The validity tests and integrals are Borel in \(V,s\).

This is one vector \(v=v(V,s)\in\mathbb R^{N_0}\) for the fixed graph and cocycle, common to all \(\omega\). Only its size and the bounds below are uniform over all graphs and cocycles of the given complexity: there are at most \(q^q\) neighborhood color assignments, \(2^q\) witness cliques, and \((C/r)^{Cq}\) projected indices, so \(N_0\) is elementarily bounded in \(q\). An actual cell receives the slot \(t(C,\omega)\) obtained by normalizing its canonical anchor, restricting its actual coloring, and choosing the first valid witness clique in the fixed order. Its value is \(u_C=v_{t(C,\omega)}\). This type is Borel and is unchanged by translating \(C\) and \(\omega\) together. Slots with the same anchor, restricted coloring, axes, and grid index but different valid witness cliques have the same value: the \(5\delta\) observation gives the identical active-label formula throughout the primal cell.

Impose on this list every integer row \(b\) with \(\|b\|_\infty\leq2(n+1)\) for which \(b\cdot v=0\). There are finitely many candidate rows, so the resulting matrix depends Borelly on \(v\). In particular it includes the difference row for any two duplicate valid slots. It also includes every needed cubical equation, even when its faces have anchors normalized by different deck elements: all their values are entries of this same vector \(v\), and Stokes gives the signed zero sum of those entries. After repeated slots in its \(2(n+1)\) faces have been combined, each coefficient has the stated bound. The kernel of this integer matrix has a real basis of integer vectors \(b_1,\ldots,b_k\) with \[k\leq N_0,\qquad \max_j\|b_j\|_\infty\leq K_0:=(C_nN_0)^{C N_0}.\] To see this, choose a nonsingular maximal minor and solve for its pivot coordinates in terms of the free coordinates; multiplying each basis vector by its nonzero determinant makes it integral. Hadamard’s bound for minors of a matrix with entries at most \(2(n+1)\) gives the displayed estimate. Lexicographically choosing the pivot minor makes the basis Borel. The case of a zero-dimensional kernel simply has \(v=0\) and needs no rounding.

Write \(v=\sum_j t_jb_j\), using the pivot solution, which is Borel. Independently round each \(t_j\) to the two adjacent multiples of \(D_0^{-1}\), choosing their probabilities so that its conditional expectation is \(t_j\). Choose a single integer \(D_0\geq N_0K_0r^{-n}q^n\), using uniform elementary upper bounds for \(N_0,K_0\). The resulting vector \(v'\) lies in the same kernel, has entries in \(D_0^{-1}\mathbb Z\), has expectation \(v\), and satisfies \(\|v'-v\|_\infty\leq N_0K_0/D_0\leq r^nq^{-n}\). All operations, including taking floors and using independent uniform parameters for the adjacent choices, are Borel. The fixed slot order is used for every pivot choice. Thus one parameter \(\xi\), independent of \(\Omega\), gives one rounded vector \(v'(V,s,\xi)\) for all cells. Set \(u'_C=v'_{t(C,\omega)}\). The invariance of the type under translation proves equivariance, and the common zero rows prove all the asserted incidence equations. ◻

Proof. Invariant reflection states. For each nonempty proper subset \(S\subset\{0,\ldots,m\}\) and each face with exactly those vertex colors, pair the incident positive chambers with the incident negative chambers. The two counts agree. Indeed choose a color omitted by \(S\); the given codimension-one matching in \(\mathcal P\) across facets omitting this color pairs the chambers at that face with opposite signs. All incident sets are finite by the incidence bound above. Choose uniformly among their possible pairings, independently for each face and each \(S\), and let \(p_S\) denote the resulting involution on chambers. Use also independent uniform bits \(t_S\in\{0,1\}\), one for each \(S\). The state space consists of \(\zeta=(\sigma,p,t)\), with counting measure on \(\sigma\) and these product probabilities on \(p,t\). These are countable products of finite probability spaces. As the original data vary, the incidence lists and finite pairing choices are Borel, so the product laws are Borel probability kernels.

Define a move \(s_S\) by \[\sigma\longmapsto p_S\sigma,\qquad t_S\longmapsto1-t_S, \qquad p_R\longmapsto p_Sp_Rp_S\quad(R\subsetneq S),\] leaving the other data unchanged. The conjugated \(p_R\) is again a pairing at each \(R\)-face: \(p_S\) preserves every such face and flips the chamber sign, so conjugation preserves its face and its sign-reversing property. Each \(s_S\) is an involution. If \(R\subsetneq S\), the two moves commute. On chambers, for example, their two orders both give \(p_Sp_R\sigma\), since after \(s_S\) the \(R\)-pairing is \(p_Sp_Rp_S\). On a pairing indexed by a subset of \(R\), both orders conjugate by \(p_Sp_R\); on the other indices the same assertion follows directly from which of the two moves changes that index. The bits commute as well. Thus all moves in a nested flag commute, and they act freely: each changes its own bit, so no nonempty product of these moves can fix a state.

These moves preserve the state measure. They permute the counting chamber index, flip a uniform bit, and, conditional on \(p_S\), conjugate each relevant uniformly distributed finite pairing by a permutation of its incident set. The latter operation permutes its finite set of possible pairings and preserves its uniform law. It also preserves the independent product of those laws. All moves are Borel and commute with translating the original label data.

Smooth leaves and the measured atlas. Attach the following tile to each state. Start with the positive simplex \(x_0,\ldots,x_m>0\), \(\sum_i x_i=1\), and take the closure of its collection of pair ratios \((x_i/x_j)_{i\ne j}\) in the product of copies of \([0,\infty]\). For each permutation \(i_0,\ldots,i_m\) use the coordinate chart \[ u_j=x_{i_j}/x_{i_{j-1}}\in[0,4),\qquad 1\leq j\leq m. \tag{152}\] All pair ratios in the chart are products of these or their reciprocals, so the \(u_j\) are independent coordinates. Sorted orders have \(u_j\leq1\) and cover the closure. The facet \(u_j=0\) has label \(S_j=\{i_0,\ldots,i_{j-1}\}\); the labels of a corner are nested. Glue each facet to the identical resolved coordinates in the state obtained by \(s_{S_j}\). The commutation just proved makes the gluing consistent at corners.

Use signed square root coordinates \(u_j=Z_j^2\), with \(Z\in(-2,2)^m\), starting from any state. For \(Z_j<0\) use the tile obtained by \(s_{S_j}\); use the product of these commuting moves when several coordinates are negative. At zero use the starting tile as a convention for representing the same glued point.

Figure [fig:resolution] illustrates the tile and the four signed quadrants at a two-facet corner when \(m=2\).

We verify that these are ordinary smooth charts. On an overlap of two permutations, a cut \(S_j\) which is not a cut of the other permutation has \(u_j\geq4^{-2m}\). To see this, choose a pair of indices on opposite sides of that cut whose order is inverted in the other permutation. Their ratio in the first ordering is at most \(4^m u_j\); its inverse in the second ordering is at most \(4^m\). The asserted lower bound follows. At a cut common to both permutations, the ratio for the second chart is the first ratio at that cut multiplied by a product of ratios within the blocks between common cuts. Those within-block ratios and their required reciprocals are bounded away from zero on the overlap. After taking square roots, the transition at a common cut is consequently its signed coordinate times a smooth nonvanishing factor. All other transition coordinates are smooth monomials with denominators bounded away from zero.

The sign in this transition is fixed on each region with fixed signs at the noncommon cuts. The remaining state relation is a product of the common reflections, and its product is unique by their free action on the bits. Commutation makes this sign rule persist across the zero coordinates. Thus the transitions are smooth and invertible across every corner. A point incident to \(j\) facets has exactly the finite star of \(2^j\) states under these reflections, and hence has one neighborhood of this form rather than several branches.

The signed charts give smooth leaf neighborhoods. We next check the topological and transverse properties needed for measured integration. The leaves are Hausdorff. Distinct points with distinct resolved coordinates can be separated in the tile model. If their resolved coordinates agree but their finite stars differ, choose neighborhoods in these two stars avoiding all nonincident facets; no further identification then joins them. Each leaf has at most countably many states reachable by the finitely many moves and at most countably many charts, so it is second countable. The total space is standard Borel: the quotient of the state–tile space identifies finite classes at each tile point, for which Borel representatives can be selected by a fixed enumeration. More explicitly, charts can be indexed by the starting chamber, its bit array, and the flag, with the pairing arrays as transversal parameters. The bits determine the crossings at nonzero signed coordinates; the moves can then be inverted to recover the pairing array. With the zero convention above this makes the chart maps injective. Changes of transversal are the measure-preserving moves on countably many Borel overlap pieces. A plaque meets only elementarily many plaques in this atlas: for each second flag only the states in the two finite flag reflection groups can occur. These observations give precisely the measured flow-box properties of Theorem 16.

The chamber signs orient the leaves. In a flag chart the Jacobian sign from positive ratio coordinates to oriented barycentric coordinates depends only on the permutation. Changing one signed coordinate from positive to negative changes the Jacobian sign of \(u_j=Z_j^2\) once, while the move changes the chamber sign once. The induced orientation is therefore the same across each facet. The transverse measures here are nonnegative; the signs enter only through this orientation in integrals of forms. The value of \(\omega\) is constant along each tile leaf, and translation by \(\Gamma\) reindexes the states and preserves these measures.

The map and the local current. Recover the raw \(x_i\) from (152) by normalizing the successive products \(1,u_1,u_1u_2,\ldots,u_1\cdots u_m\). If \(v_{\sigma,i}\) are the vertices of the indicated chamber in its flat chart, map its tile by \[ X=\sum_{i=0}^m b_i v_{\sigma,i},\qquad b_i=\frac{\exp(-1/x_i)}{\sum_l\exp(-1/x_l)}, \tag{153}\] where \(\exp(-1/x_i)\) is extended by zero at \(x_i=0\). On a facet with label \(S\), all coefficients outside \(S\) vanish to infinite order in its signed coordinate. The vertices in \(S\) are the same on the paired chambers, and their coefficients are smooth functions of the squares of the signed coordinates. Thus \(X\) is smooth through the gluing. Its denominator is at least \(e^{-(m+1)}\), since some \(x_i\geq1/(m+1)\), which also gives elementary upper derivative bounds in the tile charts.

On the positive simplex the map \(x\mapsto b\) is a diffeomorphism. Given positive \(b_i\) of sum one, its inverse is obtained from the unique \(t\in(0,1/\max_i b_i)\) satisfying \[\sum_i\frac1{-\log(tb_i)}=1;\] the left side increases smoothly from zero to infinity. The inverse function theorem also gives the smooth inverse. This diffeomorphism preserves orientation. Indeed it extends to the closed simplex preserving every face, and the linear homotopy to the identity has the same property, so its relative topological degree is \(+1\). It follows that the tile interior maps diffeomorphically, with sign \(\varepsilon_\sigma\), onto the interior of its chamber. Each chamber’s state probability integrates to one and the tile boundaries have leafwise measure zero. Fubini and change of variables therefore give, for every compactly supported smooth \(m\)-form \(\alpha\), \[\int_{\mathcal A}X^*\alpha =\sum_\sigma\varepsilon_\sigma\int_\sigma\alpha.\] This proves (150). To obtain the local identity, define \(\mathcal A_{\rm orig}\) as the image in \(X^{-1}(\mathcal T_\delta)\) of tile points belonging to original chambers. At an interior point this subset is leaf-open. At a lower face, the incidence argument preceding the lemma shows that every state in its finite reflection star is on the same side. A neighborhood in that star can be chosen to keep its image in the strict tube and to avoid nonincident facets, so it stays on that side. This proves leaf-openness and makes the definition unambiguous at glued points; the finite indexed charts also make it Borel. Applying the same change-of-variables calculation only to original tile interiors, for a form supported in \(\mathcal T_\delta\), gives (149): each original cube is counted with its coefficient \(D_0u'_C\), and each chamber’s state fiber has mass one.

The metric and upper density. The signed charts admit a background metric \(k_0\) with elementary bounds for its coefficients, inverse, and every fixed number of derivatives in buffered charts. To construct it, take smooth product cutoffs supported in the signed boxes \((-2,2)^m\) and equal to one on \([-1,1]^m\), and average their Euclidean chart metrics with the normalized cutoff weights. The smaller boxes cover, by sorted flags. The overlap bound above bounds the number of terms per plaque. The transition formulas and their inverses are products of at most \(O(m)\) factors, and every required reciprocal at a noncommon cut is at most \(4^{O(m)}\). Their fixed-order derivatives consequently have elementary bounds. These facts give the claimed bounds for the average and its inverse, and make the construction Borel and equivariant. The same transition and product estimates, applied to (153), give an elementary upper bound for \(dX\) and its fixed-order derivatives before any choice of \(\lambda\). The affine simplexes have diameter at most \(r\sqrt m\).

Put \(\Theta=\prod_i x_i\). Choose a smooth \(j=j(\Theta)\) equal to one for \(\Theta\leq\gamma_0\), zero for \(\Theta\geq2\gamma_0\), and between zero and one elsewhere, with the rescaled smooth cutoff estimates. It is smooth in the signed charts and the same on both sides of every facet. Set \[ k=X^*k_{\rm flat}+j^2 k_0, \qquad \text{seam}=\{\Theta\leq2\gamma_0\}. \tag{154}\] We call the complement of this closed seam the straight part. The metric is positive definite near the tile boundary because \(j=1\) there. Where \(j<1\), one has \(\Theta>\gamma_0\), hence \(x_i\geq\gamma_0\) for every \(i\), and \(X\) is an immersion. The formula proves \(\|dX\|_k\leq1\) and equality of metrics off the seam.

The upper density bound requires no inverse estimates. As quadratic forms, \(k\leq X^*k_{\rm flat}+k_0\), so its volume density in every sorted box is bounded by the upper bounds for \(dX,k_0\) already chosen. Call a common elementary bound \(H_0(q)\). In a sorted box \(|Z_j|\leq1\). If all \(|Z_j|>\lambda\), normalization of the successive products gives \[\Theta\geq(m+1)^{-(m+1)} \lambda^{2(1+2+\cdots+m)} =(m+1)^{-(m+1)}\lambda^{m(m+1)}.\] Condition (151) shows that the seam lies in the stated union of slabs. Its coordinate volume in such a box is at most an elementary factor times \(\lambda\). The bound \(H_0(q)\) was fixed before this factor was made small.

The remaining metric estimates may depend on \(\gamma_0\). On the part with \(j<1\), all positive ratios are bounded below by powers of \(\gamma_0\), so the product formulas control the inverse from raw barycentric to signed coordinates. The normalized weights in (153) obey \(b_i\geq e^{-1/\gamma_0}/(m+1)\). On a tangent vector \(w=(w_i)\), \(\sum_i w_i=0\), their derivative is \[db_i(w)=b_i\left(\frac{w_i}{x_i^2} -\sum_l b_l\frac{w_l}{x_l^2}\right).\] These differences control \(w\): if their common subtracted mean is \(a\), the relation \(\sum_iw_i=0\) expresses \(a\) as a weighted average, with weights \(x_i^2\), of the negatives of those differences. Thus \(|w|\) is bounded by a polynomial in \(m\) times \(e^{1/\gamma_0}\max_i|db_i(w)|\). Finally, the affine map from the barycentric simplex to a simplex in a subdivided cube has elementary inverse bounds. Its matrix has \(r\)-scaled rational entries with bounded denominators, and its nonzero determinant and cofactors give this by minors. These estimates control the inverse metric where \(j<1\); where \(j=1\), \(k_0\) already supplies that control. Differentiating the formula for \(k\) a fixed number of times now gives elementary bounds, including the curvature bound, when \(\gamma_0^{-1}\) is elementary.

Frame interpolation. The comparison needs a constant frame ordering tangent and normal axes on each straight tile, a frame that glues smoothly near facets, and a homotopy to the identity for the degree calculation. Choose a Borel skew-symmetric logarithm \(A_\sigma\) of each \(O_\sigma\) with operator norm at most \(\pi\). Such a choice follows by decomposing an orthogonal rotation into its two-dimensional angle blocks with principal angles and, on its \(-1\)-eigenspace, pairing a Borel choice of orthonormal basis; the even dimension there follows from determinant one. Define on that tile \[\mathcal R=\exp((1-j)A_\sigma).\] It is the identity in a neighborhood of each facet and thus glues smoothly despite the change of \(\sigma\). It equals \(O_\sigma\) where \(j=0\), and replacing \(1-j\) by \(t(1-j)\), \(0\leq t\leq1\), gives the asserted homotopy. The cutoff and metric inverse estimates give the derivative bounds. The choices can be made on the indexed chamber data, so they are Borel and equivariant as asserted. ◻

Proof. The local pushforward (149) will express the degree using a smoothed form of \(Z_{\rm rd}\). Its primal-cell periods are the integer coefficients \(D_0u'_C\). Their expectations match the periods of \(D_0\eta\); a compatible piecewise cubical primitive and mollification then give the same measured pairing with \(F\). We first verify the band control that makes these integrations valid.

The controlled comparison band. Before taking the quotient, the product flow boxes use the entropy transversal \(\omega\) and the state transversals of \(\mathcal A\). Variation along the tile keeps \(\omega\) constant. These transversals carry the invariant measures constructed above. A group element that stabilizes a product leaf acts freely and properly because it acts so on the \(\widetilde M\) factor. Hence the quotient leaves are ordinary oriented Hausdorff second countable manifolds. The actions reindex labels and states, leave the color and flag coordinates unchanged, and preserve the transverse measures and orientations.

For an inner closed comparison cube there is a uniform margin from the boundary of the allowed sheet domain. Indeed \(\mathcal R\) is orthogonal and (155) gives \(|z_F-z_X|_2\leq3\tau\sqrt q\ll\delta/4\). The large witness color at \(F\) remains present with a uniform small buffer. In a flat ball chart a lifted path whose projected coordinates stay in the ball stays in that same lift, by uniqueness of lifts for a local diffeomorphism. The nearby points therefore remain in coherent sheet charts. Facets that lead to the mirror are outside this region. The available smooth bounds for the instance give fixed buffered coordinate charts for all the data in these regions.

To check the finiteness condition on those charts, choose lifts of a finite buffered atlas on \(M\). On their compact closures \(F\) uses only uniformly finitely many possible labels, by the lifted support bounds in Theorem 9. Any nearby cell lift has its anchor in a closed one-neighborhood of one of these labels. The numbers of projected grid indices, chamber subdivisions, and copies there are bounded as above. Thus only finitely many chamber indices are needed over each such base chart, uniformly for the instance. Their state probabilities have total mass one per chamber, and the sum of transverse measures of this covering is finite. Plaque overlaps are bounded by the tile overlap bound and the bounded overlap of the base atlas. In the quotient only finitely many deck changes meet two of these buffered base charts. Subdividing their inner boxes and, when necessary, restricting to Borel transverse sets gives the flow boxes stipulated in Theorem 16. All metric and data bounds in this verification may depend on the instance. It also justifies Fubini and measured Stokes for the integrations that follow.

The local convolution form. Choose a smooth unit-integral top form \(\beta\) on \(\mathbb R^q\) supported in a Euclidean ball about zero of radius less than \(\min\{\tau,r/10\}\). In calculating the degree using \(\beta\), homotope \(\mathcal R\) to the identity as in Lemma 21. All maps in this homotopy are orthogonal on the difference variable. Their pullbacks of \(\beta\) are thus supported where the difference has this same small norm, strictly inside both the sheet domain and the original band. Measured Stokes for this supported homotopy shows that it does not change the degree.

After removing the rotation, fiber integration of \((z,w)\mapsto z-w\) against \(\beta\) and the original current \(Z_{\rm rd}\) gives a smooth \(n\)-form \(\alpha_{\rm rd}\) near \(K\). More explicitly, in a flat chart put \(\mathsf d(z,w)=z-w\). Integrate \(\mathsf d^*\beta\) along each compatible \(C^\#\) in the second variable, multiply by \(D_0u'_C\), and sum. The sum is locally finite and uses the original side. By the local identity (149), Fubini, and change of variables on tile interiors, the degree is \[ \mathcal D=\int_{\mathcal M}F^*\alpha_{\rm rd}. \tag{158}\] Boundary cancellation of the dual cubes shows that \(d\alpha_{\rm rd}=0\) on, for example, the tube of radius \(\delta/8\) around \(K\). To verify this directly, apply fiber Stokes to each cube. Its boundary terms only involve facets within the support distance of the point \(z\); these lie in the balanced tube and cancel with their paired incidence signs.

For every primal \(n\)-cell sufficiently near \(K\) in the same sheet, \[ \int_C\alpha_{\rm rd}=D_0u'_C. \tag{159}\] Indeed a primal \(n\)-cell and a complementary dual \(m\)-cell of this grid meet only when their indices match, and then at their common center. If their axis sets differ, a coordinate fixed for both is a grid vertex for the primal cell and a half-step center for the dual cell, so their difference in that coordinate is at least \(r/2\). If their axis sets agree but their grid indices differ, a differing coordinate gives the same separation. Such pairs cannot meet the support of \(\beta\). For the matching pair, the difference map from \(C\times C^\#\) contains the support ball in its image with multiplicity one. Our fiber integration convention is \(\int_C\int_{C^\#}\mathsf d^*\beta= \int_{C\times C^\#}\mathsf d^*\beta\) with the base-first product orientation. The differential of \(\mathsf d(z,w)=z-w\) sends this ordered frame to the primal basis followed by the negative dual basis, so its orientation is positive by our choice of the dual orientation. Its integral is therefore one, proving (159). The small support and the chart buffers ensure that only the indicated lifts contribute.

The detector pairing. Set \(\alpha=\mathbb E_{\rm rd}\alpha_{\rm rd}-D_0\eta\). It is a smooth closed form on that tube, and (146), (145), and (159) give \(\int_C\alpha=0\) on all the primal \(n\)-cells there. We explain why its measured pairing with \(F\) vanishes, including when \(F\) meets grid faces.

Use all lifted full \(q\)-cells needed to cover the tube of radius \(\delta/32\) around \(K\). Taking \(r\sqrt q\) sufficiently small as above, these cells lie in the tube of radius \(\delta/8\), all their \(n\)-faces are selected, and their shared faces fit in coherent sheet charts. On one grid interval \([a,a+r]\), project a function to its linear interpolation between the endpoints and project a one-form \(f(t)dt\) to the constant one-form with the same integral. The homotopy for a one-form is the function \[\mathsf h(f(t)dt)(t)=\int_a^t \left(f(v)-r^{-1}\int_a^{a+r}f(s)\,ds\right)dv,\] which vanishes at both endpoints; the homotopy on functions is zero. Direct differentiation gives \(\mathrm{id}-\mathsf P =d\mathsf h+\mathsf h d\). Apply these operations one coordinate at a time, with the ordinary exterior differential signs. The tensor-product homotopy still commutes with tangential restriction to a face, since the one-dimensional primitive vanishes at the endpoints. The full projection of an \(n\)-form has, for each axis set, coefficients obtained by integrating on its \(n\)-faces and interpolating linearly in the remaining axes. Hence the full projection of \(\alpha\) is zero. Since \(d\alpha=0\), the homotopy supplies a primitive on each full cube, and its tangential restrictions agree on shared faces.

This piecewise primitive has bounded coefficients for the instance and is also a weak primitive on the tube: in distributional Stokes the terms on each shared face cancel by agreement of its tangential restrictions. On a smaller tube mollify it in the flat coordinates. This can be done with the same small Euclidean convolution in every chart: the buffered sheets give a unique lift of the convolution ball, and on overlaps the piecewise primitive and flat coordinates agree. Its smooth exterior derivatives are the mollifications of the smooth form \(\alpha\), and converge to \(\alpha\) there. The construction is Borel and equivariant and has bounded coefficients for the instance. Measured Stokes applied to the pullback of each smooth primitive gives zero; convergence and the uniform chart bounds give \(\int_{\mathcal M}F^*\alpha=0\). All expectations and integrals above are measurable by their locally finite chart formulas, and their absolute bounds for the instance justify their interchange. Using (144) in (158) now proves (157). ◻

Proof of Theorem 18. The case \(\|M\|=0\) is immediate. Otherwise we first prove the conditional target for the data above, choosing \(\Psi_n\) from the requirements on \(E\) derived below, and then apply it to the entropy output. Choose a positive constant \(b_*\), depending only on \(n\), whose smallness will be fixed in the estimate below, and put \[ L_T=b_*\sqrt E,\qquad L_N=b_*\sqrt{qE},\qquad h_{\rm tot}=h+L_T^2k. \tag{160}\] Once the degree hypotheses are verified, its length denominator, the product volume, and the entropy contribution will combine as \[ \frac{L_T^mE^{n/2}}{L_T^mL_N^n}\,e^{-W}dV_h =b_*^{-n}q^{-n/2}\rho. \tag{161}\] This is the cancellation of the tangent volume and all powers of \(E\). The estimates below justify the degree application and bound the straight angular sum by a dimensional multiple of \(q^{n/2}\), while the collar volume controls the seam.

Use the weight \(\phi\), constant in the tile directions. The product scalar formula and the definition of \(\mathcal S\) give \(\mathcal S(h_{\rm tot},\phi)=\mathcal S(h,\phi)+L_T^{-2}\mathop{\mathrm{Scal}}_k\). By the definition of \(K_{\rm scal}(q)\), the lower bound \(E\geq b_*^{-2}K_{\rm scal}(q)\) ensures \(L_T^{-2}\mathop{\mathrm{Scal}}_k\geq-1\). By (142), \[ \mathcal S(h_{\rm tot},\phi)\geq-C_{\rm ent}-1+2W-n\log E+E\mathfrak e. \tag{162}\]

Assign \(L_i=L_T\) to the first \(m\) comparison rows and \(L_i=L_N\) to the last \(n\) rows. On the band, the derivative in tile directions of each row of \(G\), measured with \(k\), has norm at most two. The \(dX\) term has norm at most one, and the \(d\mathcal R\) term has norm at most \(3\tau\sqrt q\|d\mathcal R\|_k\leq1\) by (155). The horizontal derivative is a row of a rotation of \(dz_F\). It follows that \[L_i|dG_i|_{h_{\rm tot}} \leq C_n\bigl(\sqrt q+\sqrt{qE}\,E^{-1/100}\bigr).\] Take \(B\) to be this upper bound increased by two. Then \(L_i|dG_i|\leq B-1=B-q/q\). The horizon condition of Theorem 16 holds after an elementary increase of \(E\): it suffices to test the smallest length \(L_T\), and, for \(E\geq1\), \[\frac{C(1+B)^{2/3}}{L_T} \leq c_{{\rm hor},n}q^{1/3}E^{-13/75} <\frac{\tau}{10q}.\] Here \(c_{{\rm hor},n}\) depends only on the absolute horizon constant and the dimensional choice of \(b_*\), and \(13/75=1/2-(2/3)(1/2-1/100)>0\). Thus the displayed strict horizon inequality follows if \(E\geq(20c_{{\rm hor},n}q^{4/3}/\tau)^{75/13}\). We may therefore apply Theorem 16 with \(d=q\).

Write \(H_i=2(L_i|dG_i|-1)_+\), as in that theorem. At a straight point in a tile, let \(\mathcal I\) be the \(n\)-set of grid axes normal to its dual cube. Put \(A_i=(Vdz_F)_i\), with covector norms measured by \(h\), and define \[ U_{\mathcal I}^2= \frac q{\mathfrak e}\sum_{i\in\mathcal I}|A_i|_h^2 \quad\text{if }\mathfrak e>0, \qquad U_{\mathcal I}=0\quad\text{if }\mathfrak e=0. \tag{163}\] Here \(\sum_i|A_i|^2=\mathfrak e\). At a straight point \(k=X^*k_{\rm flat}\) and \(\mathcal R=Q_\sigma V\) is constant. For a tangent comparison row corresponding to axis \(i\), the squared row norm is exactly \(L_T^{-2}+|A_i|^2\). Thus \[\sum_{\rm tangent}H_i \leq L_T^2\mathfrak e=b_*^2E\mathfrak e,\] by \(2(\sqrt{1+t}-1)\leq t\). Normal rows have no tile derivative, so Cauchy–Schwarz gives \[\sum_{\rm normal}H_i \leq2L_N\sum_{i\in\mathcal I}|A_i| \leq2b_*\sqrt n\sqrt{E\mathfrak e}\,U_{\mathcal I}.\] Completing the square in \(\sqrt{E\mathfrak e}\), for \(b_*<1\), shows that \[ \frac{\sum_iH_i-E\mathfrak e}{2} \leq c_n U_{\mathcal I}^2, \qquad c_n=\frac{n b_*^2}{2(1-b_*^2)}. \tag{164}\]

Choose \(b_*\) sufficiently small that \[ \mathbb E_V\exp(2c_nU_{\mathcal I}^2)\leq C_n \tag{165}\] for every fixed horizontal differential and every fixed \(\mathcal I\). Here is a direct verification of this choice. If \(u\) is one coordinate of a uniform point on \(S^{q-1}\), its sphere moments satisfy \[\mathbb E(q u^2)^j =\frac{q^j(1\cdot3\cdots(2j-1))}{q(q+2)\cdots(q+2j-2)} \leq2^j j!.\] Consequently \(\mathbb Ee^{tqu^2}\leq(1-2t)^{-1}\) for \(0\leq t<1/2\). The singular-value decomposition of \(dz_F\) expresses each \(q|A_i|^2/\mathfrak e\) as a convex combination of such random variables, with its singular-value weights. Jensen’s inequality gives the same exponential bound for that convex combination. Another application of Jensen to the sum of the \(n\) rows in (163) bounds its exponential moment by \((1-2nt)^{-1}\). Choosing, for instance, \(2nc_n<1/4\) proves (165). No independence of the rows of \(V\) is used.

At a seam point the uniform tile row bound gives a rougher estimate which is still independent of the instance. By the triangle inequality for each row, \((a-1)_+\leq a\), and Cauchy–Schwarz on the horizontal rows, \[\sum_iH_i\leq 2L_N\sqrt{q\mathfrak e} +4q L_N/L_T \leq2b_*q\sqrt{E\mathfrak e}+4q^{3/2}.\] Completing the square once more gives \[ \frac{\sum_iH_i-E\mathfrak e}{2}\leq c_{{\rm seam},n}q^2, \qquad c_{{\rm seam},n}=2+\frac{b_*^2}{2}. \tag{166}\]

Apply the degree inequality, then use (162). Its loss is \(\exp(Cq/q)\), an absolute constant. The product volume form is \(dV_{h_{\rm tot}}=L_T^m dV_h\,dV_k\), so the measure factor is exactly (161). Together with (157) and \(|\mathbb E\mathcal D|\leq\mathbb E|\mathcal D|\), this gives the following inequality, with \(C_{\rm out}=\exp(C_{\rm deg}+(C_{\rm ent}+1)/2)b_*^{-n}\), where \(C_{\rm deg}\) is the absolute exponent in Theorem 16: \[ \begin{split} D_0I(2\tau)^q\leq C_{\rm out}q^{-n/2} \int_{\mathcal M}\rho\, \mathbb E_{V,s,\rm rd} \int_{\{a:\ |G(x,a)|_\infty<b_q\}} \mathcal B(x,a)\,dV_k(a), \end{split} \tag{167}\] where the fiber integral includes its state transversals and only the compatible original-side band, and \[\mathcal B(x,a)= \begin{cases} \exp(c_nU_{\mathcal I}^2),&a\text{ straight over axes }\mathcal I,\\ \exp(c_{{\rm seam},n}q^2),&a\text{ in the seam}. \end{cases}\] The entropy constant enters only through this final prefactor, not the geometric bounds used below. The length and volume prefactor has no exponential loss in \(m\); the seam exponential will be absorbed by the choice of \(\lambda\).

Fix \(x,\omega\), and put \(y=F(x)\). At a coupled pair in (167), a fixed projected grid index has at most one possible cell lift in the ball chart of \(y\). Its full primal and dual cells fit in that chart by (155) and the buffers, and the lift there is unique. Moreover each point of its primal cell is within \(10r\sqrt q\) of \(y\): the dual point is within \(b_q\sqrt q\) of \(y\), and the dual and primal extents are at most \(r\sqrt m/2\) and \(r\sqrt n/2\). The variation choice for \(r\) and the rounding error now give the copy-count estimate \[ D_0|u'_C|\leq D_0r^n \bigl(|\eta_{\mathcal I}(y)|+2q^{-n}\bigr), \tag{168}\] where the component is in the \(V\)-rotated frame.

On a straight tile the \(k\)-volume is exactly the flat volume of its image in the chamber. The signed permutation in \(G\) does not change its cube test, which is \(|\widehat z_y-\widehat z_X|_\infty<b_q\). Summing the chamber pieces in a copy is thus bounded by the \(m\)-area of its whole dual cube inside that test; the state probabilities for each chamber integrate to one. For a fixed axis set \(\mathcal I\), summing over all projected centers and averaging the shift gives exactly \[ r^{-q}\int_{\mathbb R^q}\int_{[-r/2,r/2]^m} \mathbf 1_{\{|\widehat z_y-c-t|_\infty<b_q\}}\,dt\,dc =r^{-q}r^m(2b_q)^q. \tag{169}\] Here \(t\) is placed in the complementary axes. This identity is just the density \(r^{-q}\) of centers under uniform shift; extending the sum to the full grid is an upper bound for the selected lifted cells.

The form bound holds in every rotated frame. Using Cauchy–Schwarz jointly in \(V\) and the \(\binom qn\) axis sets, and then (165), we obtain \[ \begin{split} \mathbb E_V\sum_{|\mathcal I|=n} (|\eta_{\mathcal I}(y)|+2q^{-n})e^{c_nU_{\mathcal I}^2} &\leq \left(\mathbb E_V\sum_{\mathcal I}|\eta_{\mathcal I}(y)|^2\right)^{1/2} \left(\mathbb E_V\sum_{\mathcal I}e^{2c_nU_{\mathcal I}^2}\right)^{1/2} +C_nq^{-n}\binom qn\\ &\leq C_nq^{n/2}. \end{split} \tag{170}\] This use of Cauchy–Schwarz allows the form components and row norms to depend on the same rotation. Combining (168)–(170), the straight part of the inner expectation in (167) is at most \(C_nD_0q^{n/2}(2b_q)^q\), because \(r^{n+m-q}=1\). Multiplying by its prefactor \(q^{-n/2}\) bounds that part, for a fixed dimension-only constant \(C_{{\rm str},n}\), by \[ C_{{\rm str},n}D_0(2\tau)^q, \tag{171}\] since \((b_q/\tau)^q=(1+3/(4q))^q\leq e^{3/4}\).

For the seam use only the consequence \(|z_X-z_y|_2\leq b_q\sqrt q\) of the cube test. Discard the tile boundaries, which have leafwise measure zero. The remaining set is the disjoint union of tile interiors indexed by their actual chamber \(\sigma\) and its state arrays. Partition each such interior into positive sorted-coordinate pieces, for example by taking the first sorted permutation in a fixed order. On these pieces \(0<Z_j\leq1\), so no signed-chart reflection changes the chamber. The conditional state measure over each actual chamber has mass one, and its \(k\)-density is at most the elementary \(H_0(q)\) chosen before \(\lambda\).

Retain the compatible-sheet restriction at this point. For a fixed projected grid index the uniqueness proved before (168) allows at most one compatible geometric lift. Apply the rough copy bound \(D_0|u'_C|\leq C_{\eta,n}D_0r^n\) from that equation, where \(C_{\eta,n}\) depends only on the form norm, and use the uniform density bound. Only after these bounds may we drop the restrictions on available sheets, roundings, and chambers and sum over all projected centers and chamber shapes.

For a fixed actual chamber shape, normal axis set \(\mathcal I\), and positive sorted flag, (153) has rotated position \[\widehat z_X=s+r\left(k+\tfrac12\mathbf 1_{\mathcal I}\right) +d_{\mathcal I,\mathrm{shape},\mathrm{flag}}(Z).\] Here \(\mathbf 1_{\mathcal I}\) is the coordinate indicator vector of \(\mathcal I\). The displacement \(d\) is independent of the pairing and bit arrays, of \(\lambda,\gamma_0\), and of the shift; it is determined by the fixed chamber shape and the barycentric formula. For each fixed \(Z\) the shift integral of the rough test is consequently \[\begin{split} \frac1{r^q}\int_{[0,r)^q}\sum_{k\in\mathbb Z^q} \mathbf 1_{\{|s+r(k+\frac12\mathbf 1_{\mathcal I})+d(Z)-\widehat z_y|_2 \leq b_q\sqrt q\}}\,ds &=r^{-q}\mathop{\mathrm{Vol}}_{\mathbb R^q}(B_{b_q\sqrt q})\\ &\leq r^{-q}(4\tau\sqrt q)^q. \end{split}\] This calculation is valid despite the shift dependence of the actual diagram, because the copy, lift, and density bounds were imposed before the enlargement to all centers. Tonelli’s theorem applies to these nonnegative bounds.

In each positive sorted piece the seam lies in \(\bigcup_j\{Z_j\leq\lambda\}\), of coordinate volume at most \(m\lambda\). A cube has \(2^m m!\) barycentric top chambers and at most \((m+1)!\) sorted flags per chamber. Thus the seam part of the inner expectation is bounded by \[ C_*(q)\lambda D_0(2\tau)^q, \tag{172}\] where one may take the elementary envelope \[C_*(q)=\max\left\{1,\, C_{\eta,n}H_0(q)\binom qn(2^m m!)(m+1)!\,m\, r^{n-q}(2\sqrt q)^q\exp(c_{{\rm seam},n}q^2)\right\}.\] Every factor here was bounded before choosing \(\lambda\). Thus \(C_*\) is independent of \(\lambda,\gamma_0,\tau\), of the bounds \(K_R,K_{\rm scal}\), and of \(C_{\rm ent}\). The powers \(\tau^q\) in the shift count have already canceled those in \((2\tau)^q\).

The two fiber estimates now have the same normalization. Insert (171) and (172) into (167), using \(q^{-n/2}\leq1\) for the seam, and cancel the positive factor \(D_0(2\tau)^q\). For the conditional data and parameter bounds used so far, this gives \[ I\leq C_{\rm out}\bigl(C_{{\rm str},n}+C_*(q)\lambda\bigr) \int_{\mathcal M}\rho. \tag{173}\]

We now fix these bounds before invoking the entropy theorem. For each integer \(q\geq n+1\), the form derivative bound first gives a deterministic \(r(q)\) with elementary reciprocal. The size bound for the universal type list, \(D_0\), incidence counts, and the bounds for \(k_0,dX\), including \(H_0(q)\), have elementary envelopes uniform over all permitted graphs and cocycles. Fix \(b_*\) using (165); this determines the dimensional constants \(c_{{\rm seam},n}\) and \(c_{{\rm hor},n}\), independently of \(C_{\rm ent}\). The displayed formula for \(C_*(q)\) is consequently available at this stage. Set \[\lambda(q)=\frac1{4C_*(q)},\qquad \gamma_0(q)=\frac14(m+1)^{-(m+1)}\lambda(q)^{m(m+1)}.\] These deterministic choices have elementary reciprocals and satisfy (151). The resolution estimates now give common elementary envelopes for its fixed-order chart derivatives, hence \(K_R(q),K_{\rm scal}(q)\), and then the common \(\tau(q)\) in (155). None of these choices uses the entropy constant.

For integer \(j\geq n+1\), define \[\Psi_n^\circ(j)=\max\left\{1,\ b_*^{-2}K_{\rm scal}(j), \left(\frac{20c_{{\rm hor},n}j^{4/3}}{\tau(j)}\right)^{75/13} \right\},\] and define a majorant on all \(t\in[1,\infty)\) by \[\Psi_n(t)=1+\max_{\substack{j\in\mathbb Z\\ n+1\leq j\leq\max\{n+1,\lceil t\rceil\}}} \Psi_n^\circ(j).\] Every expression used here has an elementary upper bound; the finite maximum is bounded by an increasing elementary envelope at \(\max\{n+1,\lceil t\rceil\}\). Thus \(\Psi_n\) is an elementary majorant fixed solely from \(n\), and it dominates both the curvature and strict horizon requirements. Hence every conditional datum with \(E\geq\Psi_n(q)\) satisfies all hypotheses used in (173). Our choice \(C_*(q)\lambda(q)=1/4\) and the definition of \(C_{\rm out}\) prove (143) with \[K_n=e^{C_{\rm deg}+1/2}b_*^{-n} \bigl(C_{{\rm str},n}+1/4\bigr).\] This closes the conditional target for all the specified data before any entropy family is chosen.

Now apply Theorem 9 with \(\mathcal B=\Psi_n\), \(E_{\min}=1\), the detector root map as \(F^{\mathrm{in}}\), and the one-point transverse space \(\Omega_0\). The detector construction uses finitely many compactly supported bump representatives and their deck translates, giving the required uniform support and derivative bounds in a lifted atlas and the equivariant compact support sets. After pulling the detector map to \(\Omega\) and embedding its labels equivariantly among the output’s finite copies, the measured affine probability homotopy of Section 2 preserves \(I\); the output therefore supplies the conditional data with \(E>\Psi_n(q)\). Its constant \(C_{\rm ent}=C_{\rm ent}(n,\Psi_n)\) depends only on \(n\). Since \(\rho\leq dV_g\) and the transverse measure over \(M\) is a probability measure, \[I\leq K_ne^{C_{\rm ent}/2}\int_{\mathcal M}\rho \leq K_ne^{C_{\rm ent}/2}\mathop{\mathrm{Vol}}_g(M).\] Finally \(I\geq\|M\|/2\), so the constant \(C_n=2K_ne^{C_{\rm ent}/2}\) proves the theorem. ◻

Proof of Theorem 1. Fix \(n\geq3\), and let \(C_n>0\) be the constant for the normalized scalar-curvature estimate in Theorem 18. Proposition 3 gives \[\|M\|\leq C_n\int_M(\mathop{\mathrm{Scal}}_g^-)^{n/2}\,dV_g\] for every metric in the theorem. Taking \(a_n=C_n^{-1}>0\) proves the stated integral bound. ◻