Proof. The bounded-cohomology mapping theorem identifies real bounded cohomology of \(M\) with that of \(B\Gamma\) by pullback along \(c_M\). The duality between bounded cohomology and the \(\ell^1\)-seminorm implies that \(\|M\|>0\) supplies a bounded class whose ordinary cohomology image evaluates nontrivially on \([M]\). These are the mapping and duality results of (Gromov 1982); neither requires a spin structure.

If \((c_M)_*[M]\) vanished rationally, its real image would vanish. Every bounded class pulled back from \(B\Gamma\) would then evaluate trivially on \([M]\), a contradiction. In dimension one, real bounded cohomology already vanishes and the assertion is vacuous; the direct circle argument below also suffices. ◻

Proof. Run the short-time Ricci flow starting at \(g\). Short-time existence on closed manifolds and the scalar evolution equation give \[ \partial_t\mathop{\mathrm{Scal}}=\Delta\mathop{\mathrm{Scal}}+2|\mathop{\mathrm{Ric}}|^2. \tag{2}\] See (Topping 2006, Theorem 5.2.1 and Proposition 2.5.4) for these statements in arbitrary dimension, and (Hamilton 1982) for the Ricci-flow framework. The maximum principle preserves nonnegativity. If the initial scalar curvature is positive at one point, the strong maximum principle makes it positive everywhere at any sufficiently small later time. If it is initially identically zero but the Ricci tensor is nonzero at a point, (2) gives a strictly positive initial time derivative there. At a sufficiently small positive time \(t_1\), scalar curvature is therefore positive at that point. Applying the strong maximum principle from \(t_1\) produces a metric of positive scalar curvature everywhere at a later time. Thus failure of this alternative forces \(\mathop{\mathrm{Ric}}_g=0\).

For the final assertion, suppose that \(g\) is Ricci-flat, that \(M\) is oriented, and that \(n=\dim M>0\). We use the orbit-packing argument of Milnor (Milnor 1968, Theorem 1). Its complete universal Riemannian cover \((\widetilde M,\widetilde g)\) has nonnegative Ricci curvature. Fix \(x\in\widetilde M\) and a finite generating set of \(\Gamma\). There is a constant \(D\) such that \[d(x,\gamma x)\le D|\gamma| \qquad(\gamma\in\Gamma),\] where \(|\gamma|\) is word length. Choose a positive radius \(r\) for which the balls \(B(\gamma x,r)\), \(\gamma\in\Gamma\), are disjoint. To choose \(r\), take a sufficiently small metric ball about the image of \(x\) contained in an evenly covered neighborhood. The radius-\(r\) balls about its lifts lie in distinct sheets and are therefore disjoint. All these balls have the same positive volume, say \(v_r\). The orbit balls with \(|\gamma|\le k\) lie in \(B(x,Dk+r)\). Bishop–Gromov comparison (Wei 2007, Theorem 4.1) therefore gives \[\#\{\gamma:|\gamma|\le k\}\,v_r \le \mathop{\mathrm{Vol}}B(x,Dk+r)\le \omega_n(Dk+r)^n.\] Thus \(\Gamma\) has polynomial word growth. Write \(b(k)=\#B_\Gamma(1,k)\). There are integers \(k_j\to\infty\) such that \(b(k_j+1)/b(k_j)\to1\). Otherwise, some \(\varepsilon>0\) would satisfy \(b(k+1)/b(k)\ge1+\varepsilon\) for all sufficiently large \(k\), forcing exponential growth. For each generator \(s\), \[\#\bigl(sB_\Gamma(1,k_j)\mathbin\triangle B_\Gamma(1,k_j)\bigr) \le2\bigl(b(k_j+1)-b(k_j)\bigr).\] These balls form a Følner sequence, so \(\Gamma\) is amenable. This uses only the growth bound, without a structural classification of groups of polynomial growth. Real bounded cohomology of an amenable group vanishes in positive degrees (Gromov 1982, secs. 3.0–3.1). The mapping theorem and duality used in 6 now imply \(\|M\|=0\). ◻

Proof. Choose a finite smooth triangulation of \(M\) by the Cairns–Whitehead theorem (Cairns 1934; Whitehead 1940); see also (Munkres 1966, Theorem 10.6). Label the vertex lifts equivariantly by elements of \(\Gamma\), making one arbitrary choice for each vertex downstairs. Barycentric coordinates define the labelled simplicial map \(h_0\). A smooth partition of unity subordinate to the open vertex stars, lifted equivariantly, gives \(h\); weights with the same label are added. At each point the labels that occur belong to the vertices of one lifted simplex. Cocompactness bounds their word diameters by a fixed \(s\), and it also bounds the \(\ell^1\)-operator norm of \(dh\) by a fixed \(H\). Linear interpolation of the weights stays in \(\mathcal K_s\).

For completeness, this entire homotopy is proper. If \(Q\subset \widetilde M\) is compact with \(\Gamma Q=\widetilde M\), its image during the homotopy is compact. Given a compact \(B\subset\mathcal K_s\), properness of the \(\Gamma\)-action allows only finitely many translates of that image to meet \(B\). The inverse image of \(B\) is therefore a closed subset of finitely many translates of \(Q\times[0,1]\).

Choose a class in \(H^n(B\Gamma;\mathbb Q)\) pairing nontrivially with \((c_M)_*[M]\), and represent it on the homogeneous bar resolution. On tuple chains, signed permutation averaging is an equivariant chain map lifting the identity of the augmentation. It is equivariantly chain homotopic to the identity on this free resolution. Antisymmetrizing the representative thus gives a cocycle \(\omega\) in the same class. It vanishes on repeated labels and restricts to a simplicial cocycle on every \(\mathcal K_a\).

Give the vertices of \(M\) a global order. Sending an ordered lifted simplex to its tuple of labels gives an augmentation-preserving equivariant chain map into the bar resolution. Such a map is unique up to equivariant chain homotopy: inductively, the difference in each degree is a cycle in the exact target resolution, and freeness of the source chains permits equivariant fillings. Comparison with the chain map induced by a classifying map shows that evaluation of \(\omega\) on these labelled fundamental simplices is precisely its pairing with \((c_M)_*[M]\), and hence is nonzero. ◻

Proof. A point in the realization of \(\mathcal O_L\) belongs to the interior of a unique minimal-dimensional ordered simplex. A group element fixing the point must preserve that ordered simplex, hence must fix its labels slot by slot. It is therefore the identity. This also proves freeness at points represented on faces; face identifications do not identify distinct orderings of the minimal simplex. Local finiteness and the word-metric bound make the action proper and its quotient finite. Thus \(\mathcal O_L\to\mathcal O_L/\Gamma\) is a principal cover, including when \(\Gamma\) has torsion. The quotient has a finite polyhedral subdivision. A simplex has at most \(\abs{B_\Gamma(1,L)}\) vertices after one of its vertices is translated to the identity, proving the dimension bound.

Embed the finite polyhedron \(\mathcal O_L/\Gamma\) in \(\mathbb R^{2v_0+1}\) and choose an open neighborhood retracting onto it. Choose a compact smooth domain inside that neighborhood containing the polyhedron in its interior, and double the domain. Folding the double to the domain followed by the neighborhood retraction gives a retraction from a closed oriented manifold \(C\) onto \(\mathcal O_L/\Gamma\). Its dimension satisfies the stated bound. Pull back the principal cover along this retraction. The inclusion of the polyhedron lifts to \(i\), while the pullback projection followed by \(\pi\) gives \(r\). Their composition has the required value. ◻

Proof. Use a sufficiently fine smooth triangulation, so that both primal simplices and closed dual blocks have arbitrarily small mesh. If all images over a primal simplex share a label, every pair of labels in their union is within \(2L\). This gives the first assertion.

We describe the chain construction explicitly. The regular dual-cell decomposition has closed cells that are balls and cellular subcomplexes. Subdivide it barycentrically. A simplex \(s'\) of this subdivision is a flag of dual cells; let \(B(s')\) be the closure of its largest cell. This is the unique minimal closed dual cell containing \(s'\). For a face \(t'\) of \(s'\) we have \(B(t')\subset B(s')\), so these closed balls form a face-compatible acyclic carrier.

There is an augmentation-preserving chain map from the subdivision chains to dual-cell chains carried by \(B\). To construct it, send a vertex to a dual zero-cell in its carrier with coefficient one. If the map has been defined below dimension \(j\), the image of the boundary of a \(j\)-simplex is an augmented cellular cycle in its carrier, because all face carriers are contained there. Fill the cycle within this contractible cellular subcomplex. Make all choices on simplex orbits and translate. The deck action on the subdivision is free, so this is equivariant. Each carrier is finite, and hence each output is finite.

Now subdivide \(\mathcal O_L\) equivariantly and approximate the actual map \(i\) closely by a simplicial map into the subdivision of the dual decomposition. Choose the target stars inside sufficiently small evenly covered balls. Simplicial approximation on the finite quotients can then be joined to the original map by a homotopy staying in those stars. Lifting that homotopy from \(i\) gives an equivariant simplicial endpoint that remains uniformly close to \(i\) upstairs. Equivalently, one may choose the stars directly on free vertex orbits. Compose source subdivision, this simplicial chain map, and the carrier map just constructed. This defines \(\mathcal F\). Contractibility of \(\mathcal O_L\) is not needed: the only fillings take place inside the target carrier balls.

Consider a dual cell occurring on an original simplex \(e\). Its carrier lies within a small mesh error of \(i(z)\) for some \(z\in e\). The primal simplex \(\tau\) dual to that cell meets it at its barycenter; small primal mesh therefore puts every \(o\in\tau\) within another small error of such an \(i(z)\). Choose all meshes and the approximation close enough to apply the common-label property to the pair \(o,i(z)\). A common label \(v\) belongs both to \(r(o)\) and to \(r(i(z))\). Since \(ri=\pi\), the latter support consists of vertices of \(e\). For a label \(\gamma\) of \(r(o)\) and a vertex \(w\) of \(e\), we obtain \[d_\Gamma(\gamma,w) \le d_\Gamma(\gamma,v)+d_\Gamma(v,w)\le L+L.\] This proves the second assertion. ◻

Proof. On an oriented \(j\)-simplex \(e=[v_0,\ldots,v_j]\) define the averaged Alexander–Whitney diagonal by \[\Delta_{\mathrm{AW}}e =\frac1{(j+1)!}\sum_{\varpi\in S_{j+1}}\mathop{\mathrm{sgn}}(\varpi) \sum_{k=0}^j [v_{\varpi(0)},\ldots,v_{\varpi(k)}] \otimes[v_{\varpi(k)},\ldots,v_{\varpi(j)}].\] Each ordering gives the usual diagonal on that simplex and its faces. Restriction of a uniformly chosen ordering to a face is again uniformly distributed, so the average commutes with the boundary. It is equivariant, and both factors in every term are faces of \(e\).

Apply \(\omega\) to the first factor of degree \(n\) and \(\mathcal F\sigma\) to the second factor, and call the resulting graded map \(T\). Its outputs are finite. In the tensor boundary, the sum of terms from the first factor vanishes after evaluation by \(\omega\) because \(\delta\omega=0\). The boundary in the second factor carries the sign \((-1)^n\), proving (7). When \(j=n\), each contributing second factor is a vertex. Its image under \(\mathcal F\sigma\) has augmentation one, while alternation makes every signed first-factor evaluation equal to \(\omega(e)\). Thus the augmentation of \(T_n(e)\) is exactly \(\omega(e)\).

Finally, every output of \(T_j(e)\) comes from a face of \(e\). By 11, labels over its corresponding primal simplex are within \(2L\) of every vertex of that face. Since \(e\) has diameter at most \(L\), they are within \(3L\) of every vertex of \(e\). ◻

Proof. Cellular duality on the oriented cover identifies finite dual chains of degree \(k\) with compactly supported primal cochains of degree \(d-k\). Write this identification as \(I\), choosing the degree signs so that \(\delta I=I\partial\). At degree zero choose the sign for which evaluation on the oriented locally finite fundamental cycle is the augmentation; the remaining degree signs are then determined successively by the boundary–coboundary identity. On product cells, define a cochain \(U\) of degree \(n+d\) by \[ \begin{split} U(e^j,e'^{\,n+d-j})&=a_j I(T(e))(e'),\\ a_n&=1,\qquad a_j=(-1)^{j+n+1}a_{j-1}\quad(j>n), \end{split} \tag{9}\] and set it to zero outside the indicated chain degrees. The cochain is invariant under simultaneous translation and thus descends to \(D_L\). The product-cell structure is valid even if the action on \(\mathcal K_L\) has inversions: the freely lifted second-factor cells give trivial stabilizers of product cells.

On a product cell of bidegree \((j,n+d+1-j)\), the two terms in \(\delta U\) are \(a_{j-1}IT(\partial e)\) and \((-1)^ja_jI\partial T(e)\). By [eq:cap-chain-map], their coefficients relative to \(IT(\partial e)\) are \(a_{j-1}\) and \((-1)^{j+n}a_j\), which cancel. Thus \(U\) is a cocycle.

Call a product cell far if all distances between vertices of its first simplex and labels of \(r\) over its closed second simplex are greater than \(4L\). This condition is inherited by faces and is translation invariant, so the far cells form a subcomplex \(D_{\mathrm{far}}\subset D_L\). The support bound in 12 shows that \(U\) vanishes on it. Let \(K_*\) be the union of the closures of all non-far cells in the quotient. There are only finitely many such cells: after fixing a second-factor simplex from finitely many orbit representatives, non-farness puts some first-factor vertex within \(4L\) of a label in a fixed finite set; the other vertices lie within \(L\) of that vertex. Local finiteness of the word metric gives only finitely many choices.

The same argument gives the required support bound. One pair of labels in a non-far product is within \(4L\), the first simplex has diameter at most \(L\), and the union of second-factor labels has diameter at most \(2L\). Every pair in its closure is consequently within \(4L+L+2L=7L\). Furthermore \(D_L\setminus K_*\subset D_{\mathrm{far}}\), so the relative cellular class of \(U\) induces the required class \(u\).

It remains to compute the pairing. By proper homotopy invariance of compactly supported cohomology, replace \(h\) by \(h_0\) and write \(\Phi_0\) for the resulting graph map. On the product cell structure of \(X\), each open cell maps to an open product cell, possibly of smaller dimension. Thus \(X_{\mathrm{far}}=\Phi_0^{-1}(D_{\mathrm{far}})\) is a subcomplex. Every cell not in it lies in the compact set \(\Phi_0^{-1}(K_*)\), and there are finitely many such cells by local finiteness. The locally finite fundamental class maps successively to \[H_{n+d}(X,X\setminus\Phi_0^{-1}(K_*);\mathbb Q) \longrightarrow H_{n+d}(X,X_{\mathrm{far}};\mathbb Q).\] In the latter group it is represented by the finite product fundamental cellular chain modulo \(X_{\mathrm{far}}\). Its coefficient on each remaining top cell is the local orientation coefficient, as follows by restricting further to local homology at an interior point. Hence evaluation by the relative cellular cocycle computes the desired compact pairing.

Only bidegree \((n,d)\) contributes. For each oriented fundamental top simplex of \(M\), choose one lift and sum over all oriented fundamental top simplices of \(\widehat C\). The latter sum evaluates \(IT(e)\) on the locally finite parameter fundamental cycle and equals the augmentation of \(T(e)\), namely \(\omega(e)\). Collapsed labelled top simplices contribute zero. The total is exactly the nonzero evaluation from 9, proving [eq:compact-pairing-nonzero]. ◻

Proof. Continuity follows from the coordinate definition and local finiteness. At any point choose a chart whose cutoff equals one, and use its \(b=1\) row. This row is the nonnegative average distance to its anchor \(g\). For \(t\in\mathcal K_L\) and any \(\gamma\in\mathop{\mathrm{supp}}t\), the triangle inequality gives \[d_\Gamma(\gamma,g) \le\sum_{\gamma'}t_{\gamma'}d_\Gamma(\gamma',g)+L \le\norm{V_o(t)}_\infty+L.\] Over finitely many trivializing charts, a bound on \(V\) therefore restricts the first-factor labels to finitely many possibilities. Closed inverse images of compact subsets are consequently compact. On \(K_*\), every input label is within \(7L+L=8L\) of the active anchor. Each distance difference in [eq:distance-profile-row] is bounded by that distance to the anchor, giving \(\norm{V}_\infty\le8L\), and hence the stated \(24L\) bound.

For the second assertion, use a chart of weight one common to both probabilities and let \(g\) be its anchor. Every label of either probability is within \(110L+s\) of \(g\). In particular a label \(\gamma\in\mathop{\mathrm{supp}}t\) equals \(gb\) for one of the rows being used. In this row the average of \(d_\Gamma(\,\cdot\,,\gamma)\) over \(t\) is at most \(s\). The row comparison implies that the average over \(t'\) is at most \(s+14R\). Since \(\mathop{\mathrm{supp}}t'\) has diameter at most \(s\), every \(\gamma'\in\mathop{\mathrm{supp}}t'\) satisfies \[d_\Gamma(\gamma',\gamma)\le14R+2s<L.\] This applies to every label of \(t\); the within-support distances already satisfy the bound.

Finally, each row of \(\mathcal V\) is locally a finite smooth sum. In a \(\widetilde M\) direction \(\sum_\gamma dh_\gamma=0\). Choose one label from the current support and subtract its distance profile before differentiating. The remaining profile values have oscillation at most \(s\), because the distance function to any fixed label is \(1\)-Lipschitz for \(d_\Gamma\). The derivative is thus bounded by \(s\abs{dh}_{\ell^1}\le sH\). A smooth nonnegative coordinate that vanishes at the point has zero differential, so zero probability coordinates introduce no additional labels into this estimate. ◻

Proof. Poincaré–Lefschetz duality for the triad \((P;E_R,E_B)\) (Hatcher 2002, Theorem 3.43) sends \(u_P\) to a class \[\alpha\in H_{q-n}(P,E_B;\mathbb Q),\] because \(\dim P=d+q\). Its relative intersection with \(j_*[N,\partial N]\) is, up to the fixed orientation convention, [eq:relative-pairing].

We spell out the relative realization using the absolute theorem of Thom (Thom 1954, Theorem III.4). Write \(k=q-n\) and clear denominators in \(\alpha\). Choose an integral relative cycle \(z\) in \((P,E_B)\) representing the resulting class. Form the double \(D=P\cup_{E_B}P'\) along the angular face. Compatible product collars at \(E_R\cap E_B\) give \(D\) the structure of a smooth compact manifold with boundary; the seam \(E_B\) is a separating hypersurface meeting the remaining boundary neatly. If the seam is empty, this double is the disjoint union of the two copies. Write \(z'\) for the reflected chain. Because the boundary chains on the seam agree, \(z-z'\) is an absolute cycle in \(D\). The restriction and excision map \[H_k(D;\mathbb Z)\longrightarrow H_k(D,P';\mathbb Z) \cong H_k(P,E_B;\mathbb Z)\] sends \([z-z']\) to \([z]\).

The compact manifold \(D\) has finite polyhedral type. Transfer the doubled class to a finite polyhedron homotopy equivalent to \(D\), apply Thom’s theorem there, and compose back to \(D\). This supplies a closed oriented smooth \(k\)-manifold \(B\) and a map \(f:B\to D\) whose fundamental class maps to a nonzero integer multiple of \([z-z']\). Reverse the orientation if necessary to make this multiple positive. Push \(f\) away from the remaining boundary of \(D\) by an inward collar deformation, chosen tangent to the seam. Smooth \(f\) and perturb it transversely to the seam. These operations are homotopies in \(D\), and compactness keeps the image away from the boundary of the seam. Thus \(Q=f^{-1}(E_B)\) is a closed smooth hypersurface, and \(B_+=f^{-1}(P)\) and \(B_-=f^{-1}(P')\) are compact oriented smooth manifolds with common boundary \(Q\). Give \(B_+\) the orientation restricted from \(B\). The map \(H_k(B;\mathbb Z)\to H_k(B,B_-;\mathbb Z)\cong H_k(B_+,Q;\mathbb Z)\) sends \([B]\) to \([B_+,Q]\): collar excision and the local orientation classes on the interior of \(B_+\) characterize this restriction. Components wholly in \(B_-\) vanish in the relative group. Naturality for the map of pairs \((B,B_-)\to(D,P')\), followed by the displayed target excision map, therefore gives \[(f|_{B_+})_*[B_+,\partial B_+] =\text{a positive integer multiple of }\alpha \quad\text{in }H_k(P,E_B;\mathbb Q).\] This proves the required relative realization without prescribing boundary data or requiring the representing map to be an embedding.

Write \(a_+:B_+\to P\) for this representative and push its image off \(E_R\) by a collar deformation respecting \(E_B\). At a corner one uses a field pointing inward from the radial side and tangent to the angular side; transversality of the two defining functions supplies such a field. Compactness now separates \(a_+(B_+)\) from \(E_R\), and already separates \(j(N)\) from \(E_B\). Consequently all coincidences of \(a_+\) and \(j\) lie in a compact subset of the source interiors, with target values in \(\operatorname{int}P\).

Perturb \(a_+\) only near this compact coincidence set. More precisely, choose finitely many target coordinate neighborhoods with closures in \(\operatorname{int}P\) and use small flows of vector fields supported there to make \(a_+\times j\) transverse to the diagonal in \(P\times P\). Compactness permits the perturbation to be small enough that it creates no coincidences outside these neighborhoods. The whole homotopy remains a map of pairs \((B_+,\partial B_+)\to(P,E_B)\), is fixed near \(\partial B_+\), and keeps the image separated from \(E_R\). Thus it preserves the relative class and its pairing with \(j_*[N,\partial N]\). The dimension sum is \((q-n)+(n+d)=q+d\), so the transverse intersections are isolated and have the asserted nonzero signed count. Set \(A_+=B_+\) and write \((o_0,y_0)\) for the perturbed map into \(E=C\times\mathbb R^q\).

Finally take a sufficiently small boundary collar, free of intersections, and precompose the map there by a smooth collar retraction that is constant in the inward collar coordinate near the boundary and is the identity away from a slightly larger collar. The image on that collar stays in a region where the strict separation inequalities hold. The resulting map is smooth, has the same interior intersections, and has the claimed collar form. ◻

Proof of 8. Double \(A_+\) along its boundary, reversing the orientation on the second copy, to obtain a closed oriented manifold \(A\). Reflect both component maps \(o_0,y_0\) across the seam. The collar conclusion of 15 makes the resulting maps smooth. Pull back \(\widehat C\to C\) by \(o_0:A\to C\), obtaining \(\widehat A\to A\), and form \[Z=X\times_C A=(\widetilde M\times\widehat A)/\Gamma.\] The projection \(X\to C\) is a submersion, so \(Z\) is a smooth \(q\)-manifold even though \(o_0\) need not be a submersion. Local sheet trivializations identify it with a product of the \(\widetilde M\) fibre directions and the \(A\) directions, with their product orientation. This also exhibits it as the stated cover of \(M\times A\). Continue to write \(\mathcal V\) for the pulled-back profile map.

Choose once and for all a smooth function \(\eta:[0,\infty)\to[0,1]\) equal to one on \([0,2]\) and zero on \([3,\infty)\), and define \[ G=\eta\bigl(\norm{\mathcal V}_{p_R}/T_0\bigr) \frac{\mathcal V-y_0}{R}. \tag{18}\] Properness of the profile map and compactness of \(A\) imply that its support is compact. It is smooth, including at \(\mathcal V=0\), because the cutoff is constant in a neighborhood of that locus. Write \(a:Z\to A\) for the parameter projection, and let \[\Omega=\{z\in Z:\ a(z)\in\operatorname{int}A_+, \norm{\mathcal V(z)}_{p_R}<2T_0\}.\] This is relatively compact. On its angular boundary the defining boundary inequality for \(A_+\) gives \(\norm{\mathcal V-y_0}_\infty>4R\). On its other boundary, \[\norm{\mathcal V-y_0}_{p_R} \ge2T_0-\norm{y_0}_{p_R}>T_0,\] and hence \(\norm{\mathcal V-y_0}_\infty>T_0/3\). The cutoff equals one on both parts of the boundary. In view of \(T_0>100L=10000R\), these bounds prove the boundary assertion in [eq:comparison-degree].

The zeros in \(\Omega\) are exactly the intersections furnished by 15: if \(\mathcal V=y_0\), their common \(p_R\)-norm is less than \(T_0\), so the point belongs to \(N\). We check orientations to identify their signed count. In oriented local lifts, the difference of the two map differentials, in tangent variables \((v_M,v_C,v_A)\) and target rows \((C,\mathbb R^q)\), is \[\begin{pmatrix} 0&I&-do_0\\ d_{\widetilde M}\mathcal V&d_C\mathcal V&-dy_0 \end{pmatrix}.\] Substitute \(v_C=v'_C+do_0(v_A)\). This substitution has determinant one. Moving the \(C\) variables to the front introduces only the fixed dimensional sign \((-1)^{nd}\), and eliminating their identity block leaves the differential of the pulled-back \(\mathcal V-y_0\) on \(Z\). Thus the intersection signs agree with its zero signs up to one fixed sign, independent of the point, sheet, and component. Division by the positive number \(R\) does not change them. Their nonzero sum is therefore \(\deg(G,\Omega,0)\) up to that sign. Doubling the parameter may cancel a global intersection count between its two copies; it does not change the local degree over the first copy used here.

It remains to check the derivative bound. By [lem:distance-profiles,eq:profile-norms], \[\abs{d_{\widetilde M}\mathcal V_i}\le K_0, \qquad \abs{d_{\widetilde M}\norm{\mathcal V}_{p_R}}\le3K_0 \quad(\mathcal V\ne0).\] In the cutoff transition region, \(\norm{\mathcal V-y_0}_\infty<4T_0\). Writing \(E_\eta=\sup\abs{\eta'}\), differentiation of [eq:comparison-map] gives for every row \[\abs{d_{\widetilde M}G_i} \le\frac{K_0}{R} +\frac{3K_0E_\eta}{T_0}\frac{4T_0}{R} \le\frac{K_0(1+12E_\eta)}{R}.\] The estimate also holds where the cutoff is constant. Together with [eq:profile-count], it proves [eq:comparison-bounds] with \(J=\max\{J_1,K_0(1+12E_\eta),1\}\), increasing \(R_0\) if necessary. Finally \(G\) is a smooth compactly supported map on the lifted product geometry of the cover \(Z\to M\times A\); each of its covariant derivatives therefore has finite supremum for this fixed \(R\). ◻

Proof. Set \[ \varepsilon=L/4,\qquad a_0=(2+\log q)^{-1},\qquad \beta=1-a_0,\qquad s=q^{-3},\qquad B_0=\frac{4L}{a_0w}. \tag{42}\] Choose a smooth nondecreasing cutoff \(\phi\) which is zero for \(t\le1/4\), one for \(t\ge1/2\), and satisfies \(0\le\phi'\le10\). Let \[ N(t)^2=s^2+(1-s^2)\phi(t). \tag{43}\] Writing \(\mathfrak p=tN'/N\), we have \[ \mathfrak p\ge0,\qquad \mathfrak p N(t)^2t^2 =\tfrac12 t^3(1-s^2)\phi'(t)\le5/8<1. \tag{44}\] The expression vanishes outside \([1/4,1/2]\), which justifies the displayed bound.

For one coordinate, denote the original function by \(\ell\) and the current full fibre metric by \(g\). Solve (32) with these choices and replace \[g\longmapsto g^+=g+N(t)^2df^2+v^2d\theta^2, \qquad \ell\longmapsto\ell^+=\varepsilon f.\] 18 applies on \(\widetilde M\times\mathbb T^b\). Its uniqueness gives torus invariance and compatibility with the parameter trivializations. Consequently \(df\) vanishes on the circle directions, so the metric retains the form (39). Its block on \(E_M\) has only increased. Since \[\abs{df}_{g^+}=vt\le1\] (the inequality is immediate for \(t<1/2\), and \(N=1\) for \(t\ge1/2\)), the new coordinate has derivative norm at most \(L/4\). Subsequent steps can only improve this bound. The tangent band gives the height estimate in (40).

It remains to estimate the scalar loss in a way that can be summed. At \(t>0\), put \(z=e_1r\). Differentiating \(\lambda(r)=B_0(r^2+s^2)^{\beta/2}\) along \(U=re_1\) gives \[\frac{U(\lambda)}{\lambda}=\gamma z, \qquad \gamma=\frac{\beta r^2}{r^2+s^2}\in[0,\beta].\] The part of \(-2U(D)\) contributed by this derivative is \(-2\gamma zD\). Hence \[ D^2+z^2-2\gamma zD \ge(1-\gamma^2)D^2\ge a_0D^2. \tag{45}\] By (44), the final term of (21) is nonnegative.

Measure the vector \(U\) in the metric at the start of the pass, and set \[b_U=\abs{U}_{g_*}.\] Since the current metric on \(E_M\) is at least \(g_*\), one has \(b_U\le r\). The input coordinate still satisfies \(\abs{d\ell}_{g_*}\le L\), so \(U(\ell)\le Lb_U\), while \(U(f)=rt\). Differentiating the tangent in the equation therefore contributes an adverse term bounded above by \[ \frac2w\left(\lambda+\frac{D^2}{\lambda}\right) (Lb_U-\varepsilon rt)_+. \tag{46}\] Since \(0\le r<1\) and \(\beta<1\), \[\lambda=B_0(r^2+s^2)^{\beta/2}\ge B_0r.\] Thus the \(D^2\) part of (46) is at most \[\frac{2L}{wB_0}D^2=\frac{a_0}{2}D^2,\] which is absorbed by (45).

For the remaining part, first suppose \(r<s\). Then \[ \lambda b_U\le \sqrt2 B_0s^{\beta+1}\le\frac{\sqrt2 B_0}{q}. \tag{47}\] Here \(s=q^{-3}\) and \(\beta>0\). If instead \(r\ge s\), then \(t\ge1/4\): for \(t<1/4\) one has \(N=s\) and \(r< s/4\). On the adverse region where \(Lb_U-\varepsilon rt>0\), it follows that \[b_U>\frac{rt}{4}\ge\frac r{16}.\] Moreover, \[r^{-a_0}\le s^{-a_0} =\exp\left(\frac{3\log q}{2+\log q}\right)\le e^3.\] Consequently, on that region, \[ \lambda b_U\le\sqrt2 B_0r^{1-a_0}b_U \le16\sqrt2 e^3B_0b_U^2. \tag{48}\] Combining these estimates and substituting \(B_0\) from (42) gives the one-coordinate inequality \[ \mathop{\mathrm{Scal}}_{g^+}\ge\mathop{\mathrm{Scal}}_g -C_0\frac{L^2}{a_0w^2}\left(\frac1q+b_U^2\right). \tag{49}\] For example \(C_0=128\sqrt2 e^3\) is sufficient. The inequality extends by continuity to the boundary of \(\{t>0\}\). On the interior of \(\{t=0\}\) the graph is constant and the update adds a constant circle, so it holds there as well.

Finally sum (49) over the \(q\) coordinates. At a fixed point of \(Z\), the inverse metrics of the \(E_M\) blocks all act on the same \(n\)-dimensional space \(E_M^*\). If \(U_i\) is the vector used at the \(i\)th step, (23) gives the telescoping identity \[ \sum_{i=1}^q U_i\otimes U_i =g_*^{-1}-(g_h^+)^{-1}\le g_*^{-1}. \tag{50}\] Taking its \(g_*\)-trace gives \(\sum_i b_{U_i}^2\le n\). The \(q\) terms \(1/q\) sum to \(1\), proving (41). In particular the growing number of old circle directions does not appear in this trace. Uniform control and smooth parameter dependence at the new stage follow from 18 applied successively a finite number of times. ◻

Proof. In a parameter chart about \(a_*\) write \(a=a_*+P^{-1}y\). The parameter block becomes \(g_A(a_*+P^{-1}y)\), and the other metric coefficients are the corresponding fibre coefficients evaluated at \(a_*+P^{-1}y\). All parameter derivatives through order two tend uniformly to zero. Their limits, at \(y=0\), are the two-jets of the fibre metric with \(a=a_*\) frozen, times the constant Euclidean parameter block \(g_A(a_*)\). Inverses remain uniformly bounded for this fixed stage. The scalar-curvature formula is a continuous expression in these metric two-jets and their inverses, so its limit is uniformly the fibre scalar curvature. The finitely many parameter charts and the assumed bounds make the convergence uniform over all of \(Z\).

The parameter differential of \(H\) acquires the factor \(P^{-1}\), proving the second assertion. The reference product on \(Z\) is the pullback of a metric on the closed manifold \(M\times A\) and is complete. The assembled metric, with its finite number of circle blocks, is uniformly equivalent to this complete reference product and has bounded coefficient derivatives. ◻

Proof. Start with the lifted metric of \(M\) and no circles, and perform passes indexed by \(j=0,\ldots,k-1\) with \[ L_j=4^{-j}L_0,\qquad w_j=2^{-j}/10. \tag{53}\] 19 applies inductively because every coordinate derivative is divided by at least four in each pass. Every stage is finite, so the uniform control required for its successor follows from the last assertion of that Lemma. All local constructions match under the flat parameter transitions by uniqueness of the graph solution.

For any number of passes, the total scalar loss is at most \[C_0(n+1)(2+\log q)\sum_{j\ge0}\frac{L_j^2}{w_j^2} =\frac{400}{3}C_0(n+1)(2+\log q)L_0^2.\] Choose \(C_n=(400/3)C_0(n+1)\). Under (51), all the fibre scalar curvatures are at least \(3\kappa/4\). Each coordinate is modified exactly once per pass, so \[\norm{G^{(k)}-G}_{\infty} <\frac\pi2\sum_{j=0}^{k-1}w_j<\frac\pi{10}.\] No coordinate count enters this estimate. The fibrewise Euclidean operator norm is at most \(\sqrt q\,L_k\), since each of the \(q\) rows has norm at most \(L_k=4^{-k}L_0\).

At this fixed stage apply 20, making the parameter scale so large that the scalar-curvature loss from assembly is at most \(\kappa/4\) and the parameter part of \(dG^{(k)}\) has operator norm at most \(L_k\). The domain’s fibre and parameter blocks are orthogonal. Cauchy–Schwarz therefore bounds the full differential by \(\sqrt{qL_k^2+L_k^2}\), proving (52). The resulting base metric on \(Z\) is \(h_k\); all circle warps remain positive and independent of circle variables. The completeness and finite-stage bounds follow from the same assembly Lemma. ◻

Proof. The left-hand side of (51) is at most \[C_n(2+JR)J^2/R^2\longrightarrow0.\] Choose a sufficiently large finite \(R\) once. In particular \(q\), \(A\), and the original comparison map are now fixed while \(k\) remains arbitrary. Apply 21 at this fixed scale.

Choose a smooth map \(Q_q:\mathbb R^q\to S^q\) which is constant outside the Euclidean unit ball and has degree one on the one-point compactification. It can be chosen with the north pole as a regular value having the sole preimage \(0\). For example, in polar coordinates use \[y\longmapsto \left(\sin\rho(\abs y)\frac{y}{\abs y},\cos\rho(\abs y)\right),\] where \(\rho(r)\) is a positive linear function near zero, increases to \(\pi\), and is smoothly constant at \(\pi\) in a collar of \(r=1\) and beyond; choose the orientation, or compose with a reflection, to give degree one. The expression is smooth at zero because \(\rho\) is linear there. Let \(D_q<\infty\) bound its differential.

By the comparison theorem there is a precompact open \(\Omega\subset Z\) with \(\deg(G,\Omega,0)\ne0\) and \(\norm{G}_{\infty}>3\) on \(\partial\Omega\). The homotopy \[G+u(G^{(k)}-G),\qquad 0\le u\le1,\] stays outside the unit ball on \(\partial\Omega\), by (52). Define \[F_k=Q_q\circ G^{(k)}\quad\text{on }\Omega, \qquad F_k=Q_q(\infty)\quad\text{on }Z\setminus\Omega.\] The boundary is compact and its image is separated from the unit ball. Thus \(Q_q\circ G^{(k)}\) is constant on a neighborhood of the boundary, and this definition is smooth even if \(\partial\Omega\) is not smooth. The map is constant outside the compact set \(\overline\Omega\).

Homotopy invariance of local degree gives \[\deg(G^{(k)},\Omega,0)=\deg(G,\Omega,0)\ne0.\] The degree of \(F_k\) is this same integer: count preimages of a regular target sufficiently near the north pole, where \(Q_q\) is an orientation-preserving local diffeomorphism, or equivalently compose the relative degree classes. Hence \(F_k\) has nonzero degree. Its differential is bounded by \(D_q\sqrt{q+1}\,4^{-k}J/R\), establishing (54) with \(C(q)=D_q\sqrt{q+1}\).

For the last assertion, multiply each circle warp by a sufficiently large positive constant. Because all coefficients are independent of the circle variables and the circle blocks are mutually orthogonal, this is locally a constant change of each circle coordinate. It preserves scalar curvature. The base metric and the norm of \(dF_k\) are unchanged. At the fixed stage all warps have positive uniform lower bounds, so their constant multipliers can make every inverse circle length at most \(4^{-k}J/R\). The operator norm of the projection to the product flat torus is the maximum of these inverse lengths, with no factor depending on the number of circles. ◻