History and relation to earlier work

Gromov’s macroscopic dimension program relates positive scalar curvature to the dimension of a space seen at large scales. The quantitative formulation asks whether a fixed positive scalar-curvature normalization forces a dimension-dependent bound for the fibers of a map to an \((n-2)\)-dimensional polyhedron. This formulation goes back to (Gromov 1986, sec. 2.A(c)); see also (Gromov 1996§ \(2\frac12\)) and (Gromov 2023). Uniform width is stronger than merely asking for a finite fiber bound for each individual metric. The stabilized partition theorem used here is the geometric input from (OpenAI 2026) that gives the uniform conclusion.

Several earlier results illustrate the distinction between the geometric and topological versions of the problem. Early three-dimensional distance-level diameter estimates of Gromov–Lawson (Gromov and Lawson 1983) use vanishing rational first homology, as clarified in Gromov’s later account (Gromov 2023). That account also gives a uniform Urysohn \(1\)-width bound for arbitrary complete three-manifolds with scalar curvature bounded below by a positive constant. Under such a uniform lower bound, Liokumovich–Maximo (Liokumovich and Maximo 2023) and Liokumovich–Wang (Liokumovich and Wang 2026) construct Morse functions with simultaneous area and diameter bounds on connected level components, in the closed and complete noncompact cases, respectively. In dimensions four and five, Chodosh–Li–Liokumovich proved finite Urysohn \(1\)-width for universal covers of closed positive-scalar-curvature manifolds under the respective assumptions \(\pi_2=0\) and \(\pi_2=\pi_3=0\) (Chodosh et al. 2023, Theorem 4). That conclusion is qualitative and uses the topology of a compact quotient. The setting of Theorem 1 instead allows arbitrary complete manifolds and asks for one constant depending only on the dimension.

The coefficient \(4\) in (1) is natural in weighted scalar curvature. On closed manifolds, Perelman’s \(\mathcal F\)-functional and its lowest eigenvalue formulation involve \(-4\Delta+\mathop{\mathrm{Scal}}\), together with the expression \(\mathop{\mathrm{Scal}}+2\Delta t-|\mathrm dt|^2\) (Perelman 2002, secs. 1–2). Gromov’s torus-stabilized scalar curvature makes the same connection through diagonal warped products (Gromov 2024, secs. 1.A–1.B). Spectral curvature bounds also enter the band and cube inequalities of Hirsch–Kazaras–Khuri–Zhang (Hirsch et al. 2024), whose hypotheses depend on the dimension, topology, and coefficient of the scalar-curvature term. Those inequalities control separations of boundary sets, rather than the fibers of a map on an arbitrary complete manifold.

Our starting point is the partition theorem in (OpenAI 2026), including its locally finite stabilized input. The spectral-to-positive-solution passage and the warped-product formula are established tools. The additional work is to create a quadratic reserve from an unrestricted logarithmic drift and to adapt the local stretching argument to a signed drift. These constructions retain the original metric as the metric in which the final fibers are measured. In particular, no control of a conformal factor or of the weight at infinity is needed.

The new constructions and the proof

The spectral hypothesis supplies a positive smooth function \(v\) with \(-4\Delta_g v+\mathop{\mathrm{Scal}}_gv\ge v\). For a metric \(h\) and a smooth real function \(t\), write \[ \mathcal D(h,t)=\mathop{\mathrm{Scal}}_h+2\Delta_h t-|\mathrm dt|_h^2. \tag{2}\] Then \(w=-2\log v\) satisfies \(\mathcal D(g,w)\ge1\). This identity produces a weighted curvature bound, but gives no control on the value or oscillation of \(w\). The stretching construction needs an additional reserve that grows quadratically with its drift.

The first new ingredient supplies that reserve without estimating \(w\). For an arbitrary complete metric \(h\) and smooth \(w\), we construct a smooth function \(u\), a global graph metric \(h^+=h+\mathrm du^2\), and a new signed drift \(t^+\) such that \[\mathcal D(h^+,t^+)\ge\mathcal D(h,w)+(t^+)^2-C(n).\] Here \(C(n)\) is independent of the geometry and of \(w\). The graph equation has pressure proportional to the discrepancy \(u-w\); subtracting the graph height in the drift update turns that discrepancy into the required square. Local barriers and an angle estimate give a global solution even when \(w\) and the geometry are unbounded at infinity. The precise statement and proof are in Proposition 4.

The second ingredient stretches selected pairs of sets by increasing the metric on a compact neighborhood. Its pressure depends on a possibly negative drift. For a dimension-dependent \(T\), the reserve is \(V(t)=t^2+T^2\). When the drift increases from \(t\) to \(t'\ge t\), the local estimate retains the positive part of \(V(t')-V(t)\), because increasing a signed drift need not increase \(V\). Its loss is paid by a decrease of \(\mathop{\mathrm{tr}}_h h_0\), where \(h_0\) is fixed during one round of stretching, together with an arbitrarily small summable error. The inverse construction of the pressure is given in Proposition 8, and the resulting geometric estimate is Lemma 9.

To assemble these tools, choose a radius \(D\) depending only on \(n\) and maximal \(D\)-separated centers \(x_i\) in the original metric. Their closed radius-\(D\) balls cover \(M\), and their closed radius-\(2D\) balls are locally finite. The local lemma is applied in rounds until each inner ball is as far as prescribed from the exterior of its outer ball. The trace decreases telescope within each round, independently of the number of overlapping balls. Each index \(i\) requires only finitely many rounds, and every compact set meets only finitely many outer balls, so the final metric and drift are smooth and eventually unchanged on each compact neighborhood.

The final bound \(\mathcal D(h,t)\ge1/2\) is converted into the stabilized scalar-curvature inequality by expressing \(-t\) as a locally finite sum of compactly supported logarithmic warps and subdividing each summand until the sum of squared gradients is small. Theorem 2 then supplies a map whose entire fibers are bounded in \(g\). Section [assembly:section] carries out the parameter choice and this last passage. Thus the graph reserve and the signed stretching estimate are the analytic contributions; the topological dimension drop is the conclusion of the stated geometric input.

The angle equation and elliptic estimates

To construct the prescribed graph, we need an a priori gradient estimate on a possibly noncompact manifold. The useful gradient variable is the vertical component of the graph normal: a positive lower bound for this component is an upper bound for the slope. We record the equation it satisfies, allowing angle-dependent pressure for use in the later local construction. The use of normal variation and comparison to estimate graph gradients has a classical predecessor in Korevaar’s treatment of prescribed mean curvature (Korevaar 1984); we give the weighted calculation and all estimates used here.

We will use the following local elliptic facts in both graph constructions. In coordinates, the equations have the form \[ \partial_i\mathfrak A^i(x,Du)=\mathfrak B(x,u,Du), \tag{8}\] where the volume density may be included in \(\mathfrak A\) and \(\mathfrak B\). Suppose these functions are smooth on a neighborhood of a compact range of \((x,u,Du)\), with bounded derivatives there, and that \(\mathfrak A^i_{p_m}\) is uniformly elliptic on that range. For a family of equations, the stated coefficient and ellipticity bounds are assumed uniform over the parameters under consideration. Uniform height and gradient bounds for smooth solutions then give interior bounds of every order. For completeness, differentiating in \(x_j\) and writing \(b=\partial_j u\) gives \[ \partial_i\bigl(\mathfrak A^i_{p_m}\partial_m b\bigr) -\mathfrak B_{p_m}\partial_m b =\mathfrak B_u b+\mathfrak B_{x_j} -\partial_i\mathfrak A^i_{x_j}. \tag{9}\] Subscripts \(x_j\) on the coefficient functions mean explicit derivatives, with the other arguments held fixed. Thus this is a uniformly elliptic divergence equation with bounded coefficients, bounded lower-order terms, bounded scalar data, and the divergence of bounded vector data. The interior De Giorgi–Nash estimate (Gilbarg and Trudinger 2001, Theorem 8.24) gives a uniform Hölder bound for \(b\), and hence a \(C^{1,\alpha}\) bound for \(u\), for some \(\alpha>0\). The nondivergence form of (8) now has Hölder coefficients and right-hand side. Interior Schauder estimates (Gilbarg and Trudinger 2001, Theorem 6.2 and Corollary 6.3) give \(C^{2,\alpha}\) bounds, and differentiation yields bounds of all higher orders; compare also (Gilbarg and Trudinger 2001, Theorem 13.1) for this differentiation argument. On a closed manifold a finite coordinate cover makes these estimates global.

In particular, a continuation on a fixed closed manifold is closed once uniform height and gradient bounds keep the solutions in a compact subset of the open equation domain. It is open when the linearization has strictly negative zeroth-order coefficient. Indeed, for the resulting smooth uniformly elliptic operator, interpolation with \(\Delta-1\), the maximum principle, and Schauder estimates give invertibility \(C^{2,\alpha}\to C^{0,\alpha}\). The starting operator \(\Delta-1\) is invertible by the energy method and elliptic regularity. The implicit function theorem then gives openness, and elliptic regularity gives smooth solutions. All constants in this discussion may depend on the individual compact manifold or the fixed coordinate charts.

A global graph with linear discrepancy pressure

We now solve the graph equation without imposing any global size condition on \(w\). The positive derivative of the pressure with respect to height will provide both comparison and the angle bound.

Scalar comparison for a graph enlargement

The global graph is now available, and we turn to its scalar curvature. For the later localized construction we allow a more general metric which stretches only the line normal to a level set of the height function. Writing this length multiplier as \(1/l\), we again use \(t_u=t-\log l\) to preserve the weighted density. The following calculation extends the graph comparison underlying the local deformation in (OpenAI 2026, sec. 3.5, Lemma 3.4).

Identity (13) explains the role of \(M_1\): it is the weighted divergence of \(k_1e\), divided by \(k_1\). For the ordinary graph choice \(\theta=s^2/(1+s^2)\), the vector field is \(k_1e=\nabla_hu/\sqrt{1+|\mathrm du|_h^2}\). If this divergence is prescribed to be \(p\), then \(M_1=p/k_1\) and the term \(\theta M_1^2\) in (14) is precisely \(p^2\). The terms involving \(K\) and \(e(M_1)\) still have to be controlled. For the graph just constructed, the linear discrepancy pressure will control them through the following square completion.

Completion of the reserve estimate

We can now turn the discrepancy pressure into the square in Proposition 4. The argument is pointwise; consequently the compact-data constants needed to construct \(u\) will not enter the reserve constant.

The new function \(t^+\) may have either sign and need not be bounded below. Proposition 4 controls its square with a constant depending only on the dimension. The local enlargement must therefore use a pressure law that works for every real value of the drift parameter; this is the purpose of the next section.

Slope modification and spatial cutoffs

Fix the constants \[ \begin{gathered} s_0=\frac1{100},\qquad A_0=1000,\qquad P_0=20, \qquad C_* =s_0^{-1},\\ b_0=\frac{\delta s_0^2}{2000},\qquad \epsilon=\min\left\{\frac{s_0}{100},\frac1{8n}, \frac1{8C_*}\right\}. \end{gathered} \tag{35}\] Apply Proposition 8 with these parameters, obtaining \(S_0,T,F\). Throughout this section, \(N(t)=\sqrt{t^2+T^2}\) and \(V(t)=N(t)^2\). We may take \(W\) to be a numerical constant at least \(10^5\); this size allows the cutoffs below and the final length comparison.

Let \(0<a\le1/2\). This parameter will be fixed small enough in terms of the prescribed loss \(d\). Choose a smooth nondecreasing function \(\sigma:[0,\infty)\to[0,1]\) that is zero on \([0,s_0]\), one on \([5s_0,\infty)\), and satisfies \(0\le\sigma'\le60\). For \(s\ge0\) set \[ \mu=a^2+(1-a^2)\sigma(s),\qquad \theta=\frac{\mu s^2}{1+s^2},\qquad l=\sqrt{1-\theta},\quad \beta=\log l, \quad k_1=\sqrt\theta. \tag{36}\] Thus the graph modification of Lemma 7 is the ordinary graph metric when \(s\ge5s_0\), whereas its size is reduced by \(a^2\) near zero slope. For the graph equation, we use the multiplier \[j(s)=\sqrt{\theta+a^2s_0^2}.\] It is smooth as a function of the gradient covector, and \(j(0)=as_0>0\). This keeps the derivative of the pressure with respect to height positive at critical points of the graph. The same multiplier will also make the small-slope curvature error proportional to \(a^2\).

We next arrange the spatial support. Write \(r_A(x)=\mathop{\mathrm{dist}}_{h_0}(A,x)\). Choose smooth compactly supported functions \(\psi\ge0\) and \(\eta_1,\zeta\in[0,1]\) such that \[ \begin{gathered} \inf_A\psi>W/5,\qquad |\mathrm d\psi|_{h_0}\le1, \qquad \mathop{\mathrm{supp}}\psi\subset\{r_A<W/2\},\\ \eta_1=1\text{ on }\{r_A\le.55W\},\qquad \eta_1=0\text{ on }\{r_A\ge.8W\}, \qquad |\mathrm d\eta_1|_{h_0}\le s_0/8,\\ \zeta=1\text{ on }\{r_A\le.82W\},\qquad \zeta=0\text{ on }\{r_A\ge.98W\}. \end{gathered} \tag{37}\] These cutoffs require only completeness of \(h_0\). Indeed, finite-radius closed neighborhoods of the compact set \(A\) are compact. Start with Lipschitz functions of \(r_A\), leaving positive buffers around each specified constant region. In particular, smooth \((W/3-r_A)_+\) with value error less than \(W/100\) and gradient bound \(1.05\), then divide by \(1.05\) to obtain \(\psi\). The profile for \(\eta_1\) can be smoothed with gradient at most \(12/W\le s_0/8\). To carry out the smoothing, use convolutions in sufficiently small charts on the compact neighborhood, where metric distortion is as small as desired, and a partition of unity. Choose the local approximation errors small enough to control the partition derivatives. Nonnegative kernels preserve the range, and the buffers preserve the stated constant regions. The same procedure supplies \(\zeta\).

For \(0<\rho\le1/2\), put \[ \eta=\eta_\rho=\rho+(1-\rho)\eta_1^2. \tag{38}\] Thus \(\eta=1\) near \(\mathop{\mathrm{supp}}\psi\), while \(\eta=\rho\) outside \(\{r_A<.8W\}\). Moreover, since \(h\ge h_0\), \[ |\mathrm d\eta|_h\le s_0/4,\qquad \frac{|\mathrm d\eta|_h^2}{\eta}\le1. \tag{39}\] The square in (38) gives the second estimate uniformly as \(\rho\downarrow0\). For each \(\rho\) we will construct a smooth graph height \(u_\rho\) close to \(\psi\). The arrangement in Figure 1 leaves a region where the graph tends to zero before the final cutoff \(\zeta\) begins to vary.

Solving the graph equation and obtaining locality

Work for the moment on a closed auxiliary manifold extending a neighborhood of \(\{r_A\le1.1W\}\). If \(M\) is compact, use \(M\) itself. Otherwise, enclose this compact neighborhood in a relatively compact smooth domain, modify the data to product form near its boundary, and double. Extend \(h,t\) smoothly and the compactly supported cutoffs by zero. Define \(\eta\) from the extended \(\eta_1\) as above. This auxiliary manifold is fixed as \(\rho\) tends to zero; all geometric constants in this subsection may depend on it.

For \(0\le\lambda\le1\), consider the equation \[ \begin{gathered} \mathop{\mathrm{div}}_{h,t}\mathcal A(\mathrm du) =j(s)F\left(\frac{u-\lambda\psi}{\eta},t\right),\\ \mathcal A(\mathrm du)=\frac{\sqrt\mu}{\sqrt{1+s^2}}\nabla_hu, \qquad j(s)=\sqrt{\theta+a^2s_0^2}, \qquad s=|\mathrm du|_h. \end{gathered} \tag{40}\] We solve it within the open domain \(|u-\lambda\psi|/\eta<S_0\). Both \(\mathcal A\) and \(j\) are smooth functions of the covector \(\mathrm du\), including at zero. The flux derivative has tangential and radial eigenvalues \(k_1/s\) and \((k_1)_s\), respectively; both are positive for \(s>0\) and tend to \(a\) at zero.

We first obtain estimates for every smooth solution in this domain. There is \(S_1<S_0\), independent of \(\rho,\lambda\), such that \[ |u-\lambda\psi|\le S_1\eta. \tag{41}\] To see this, compare with \(\lambda\psi\pm S\eta\) for \(S<S_0\). Their first two derivatives are uniformly bounded as \(S\uparrow S_0\), whereas \(j\ge as_0\) and the poles of \(F\) are uniform on the compact range of \(t\). For \(S\) sufficiently close to \(S_0\), these functions are strict upper and lower barriers. At a positive maximum of a violation, the gradients agree, ellipticity orders the second derivatives, and \(F_x>0\) orders the right sides, a contradiction. The same fixed \(S\) therefore works for all \(\rho,\lambda\).

There is also a gradient bound independent of \(\rho,\lambda\). At a point of maximum slope, there is nothing to prove if \(s<1\). Otherwise \(\mu=1\) in a neighborhood, and (40) is the ordinary graph equation of Lemma 5, with \[p(x,z,\omega)=\sqrt{1-\omega^2+a^2s_0^2}\, F\left(\frac{z-\lambda\psi(x)}{\eta(x)},t(x)\right), \qquad \omega=(1+s^2)^{-1/2}.\] By (41), the values of \(F,F_t,F_x\) range in a compact set uniformly in \(\rho,\lambda\). Holding \(\omega\) fixed and writing \(\xi=(u-\lambda\psi)/\eta\), we have on the graph \[\partial_\nu\left(\frac{z-\lambda\psi}{\eta}\right) =\frac{\nu_z-\lambda\,\mathrm d\psi(\nu_h) -\xi\,\mathrm d\eta(\nu_h)}{\eta}.\] Thus the zeroth order coefficient of the angle operator is bounded below by \(-C/\eta\), after discarding the nonnegative term \(|\mathrm{II}|^2\). Here \(C\) is independent of \(\rho,\lambda\). On the other hand, \[p_z=\frac{jF_x}{\eta}\ge\frac{A_0T}{\sqrt2\,\eta}\] at the point under consideration. At this minimum of \(\omega\), the angle equation gives \[-p_z=\mathcal L\omega\ge-\frac{C}{\eta}\omega.\] It follows that \(\omega\) is uniformly bounded below, as required.

For each fixed \(\rho>0\), these estimates confine the solutions to compact jet ranges strictly inside the domain of the equation. The elliptic estimates described in Section 3.1 give all higher bounds. The linearization of the left side minus the right side has zeroth order coefficient \(-jF_x/\eta<0\), so it is invertible on the closed manifold by the maximum principle and linear elliptic theory. The method of continuity therefore applies from the solution \(u=0\) at \(\lambda=0\) to \(\lambda=1\). Write \(u_\rho\) for the resulting smooth solution. The higher estimates used for this continuation may depend on \(\rho\); the height and gradient estimates do not.

The graph equation has now produced a height close to \(\psi\). Its next property is what permits a compactly supported change of metric: \[ u_\rho\longrightarrow0\quad\text{in }C^3 \text{ on compact subsets of any open region where } \psi=\eta_1=0. \tag{42}\] In such a region \(\eta=\rho\). Expanding (40) in a coordinate chart gives \[ a^{ij}\partial_i\partial_j u_\rho+b^i\partial_i u_\rho -\frac{c_\rho}{\rho}u_\rho=0, \qquad c_\rho=j\int_0^1F_x(qu_\rho/\rho,t)\,\mathrm dq. \tag{43}\] The previously established gradient bound makes \((a^{ij})\) uniformly elliptic and bounded and makes \(b^i\) bounded. The first order terms have this form because the flux vanishes at zero covector. By (41) and Proposition 8, \(c_\rho\) is bounded above and below by positive constants, independently of \(\rho\).

On a coordinate box with closure in this region, let \(\ell_f\) be the inward affine distance from its face \(f\). For sufficiently small fixed \(\gamma>0\), the function \[C\sum_f\exp(-\gamma\ell_f/\sqrt\rho)\] is a supersolution of the operator in (43), for all small \(\rho\). Indeed, its second derivative terms are bounded by a constant times \(\gamma^2/\rho\), its first derivative terms by a constant times \(\gamma/\sqrt\rho\), and its negative zeroth order term dominates both. Choose \(C\) to dominate the uniform height bound on the faces and compare with \(u_\rho\) and \(-u_\rho\). This proves exponential decay of the height on every interior subbox.

To turn this into derivative decay, rescale around an interior point by \(x=x_0+\rho y\) and \(u_\rho(x)=\rho v(y)\). The equation takes the divergence form \[\partial_{y_i}\mathfrak A^i(x_0+\rho y,Dv) =\rho\,\mathfrak B(x_0+\rho y,v,Dv).\] Here \(|v|\le S_1\) by (41), while \(Dv=Du_\rho\) is uniformly bounded. All coefficient derivatives on these ranges are uniformly bounded, and ellipticity is uniform. The interior estimates from Section 3.1 give bounded derivatives of \(v\) of every order on smaller fixed balls. Returning to the original coordinates yields bounds \(C_j\rho^{1-j}\) for derivatives of order \(j\). For fixed nested subboxes \(K\subset\operatorname{int}K'\), interior interpolation gives \[\|u_\rho\|_{C^3(K)} \le C\|u_\rho\|_{C^0(K')}^{1/4} \|u_\rho\|_{C^4(K')}^{3/4} +C\|u_\rho\|_{C^0(K')}.\] The \(C^4\) norm is at most \(C\rho^{-3}\), while the \(C^0\) norm decays exponentially in \(1/\sqrt\rho\). The right side tends to zero, proving (42).

Curvature balance

For the slope functions (36), \[0\le-s\beta_s\le2.\] Indeed, \(-s\beta_s=s\theta_s/[2(1-\theta)]\); on the switching interval use \(s\le.05\) and \(\sigma'\le60\), and outside it the formula is immediate. Thus Lemma 7 applies to \(u=u_\rho\). Denote its output by \[ h_u=h+\frac{\theta}{(1-\theta)s^2}\,\mathrm du^2, \qquad t_u=t-\beta. \tag{44}\] Here \(\mathrm du^2\) denotes \(\mathrm du\otimes\mathrm du\). These expressions are smooth across \(s=0\): the coefficient of \(\mathrm du^2\) extends as \(\mu/[1+(1-\mu)s^2]\). All comparisons with \(h_0\) below take place in the original neighborhood of \(A\); no extension of \(h_0\) is needed.

On \(\{s>0\}\), put \(e=\nabla_hu/s\) and \(b=\sqrt{h_0(e,e)}\le1\). The rank-one inverse metric formula gives \[ \mathop{\mathrm{tr}}_hh_0-\mathop{\mathrm{tr}}_{h_u}h_0=\theta b^2. \tag{45}\] Use \(J,H,K,M_1\) as in Lemma 7: \(J\) is the second fundamental form of the levels of \(u\), \(H=\mathop{\mathrm{tr}}J\), \(K=-e\log k_1\), and \(M_1=H-et-K\). In particular, \[\mathcal D(h_u,t_u)-\mathcal D(h,t)\ge\theta E,\qquad E=|J|^2+M_1^2+K^2/l^2-2M_1K+2eM_1.\] Increasing the drift by \(-\beta\) has a controlled effect on the reserve: \[ \frac{(V(t-\beta)-V(t))_+}{\theta} \le4(1+s)N(t),\qquad s>0. \tag{46}\] Indeed, if \(b_*=-\beta\ge0\), the numerator is at most \(2N(t)b_*+b_*^2\). When \(\theta\le1/2\) one has \(b_*\le\theta\). When \(\theta>1/2\), one has \(s>1\) and \(b_*\le\tfrac12\log(1+s^2)\le\sqrt s\); these inequalities give (46).

Together with (45), this identifies the estimate needed at slopes \(s\ge s_0\): it suffices to prove \(E\ge4(1+s)N(t)-\delta b^2V(t_u)\). Below \(s_0\), we will instead use the adjustable small error in Lemma 9.

To express the graph equation in the normal variables, put, for \(s>0\), \[ \chi=\frac{j}{k_1}=\sqrt{1+\frac{a^2s_0^2}{\theta}},\qquad m_1=\frac{\mathrm d\log\chi}{\mathrm d\log k_1} =-\frac{a^2s_0^2}{\theta+a^2s_0^2}. \tag{47}\] The factor \(\chi\) normalizes the pressure after division by \(k_1\), and \(m_1\) records its derivative. Directly from these formulas, \[ \begin{gathered} -1\le m_1\le0,\qquad 1\le\chi\le2\quad(s\ge s_0),\\ m_1\le-\tfrac12,\qquad \chi\ge s_0/s,\qquad \theta\chi^2\le2a^2s_0^2\quad(0<s\le s_0). \end{gathered} \tag{48}\] For \(s\le s_0\), substitute \(\theta=a^2s^2/(1+s^2)\); for \(s\ge s_0\), use \(\theta\ge a^2s^2/(1+s^2)\) to bound \(\chi\).

Equation (40) at \(\lambda=1\) says \(M_1=\chi F\), with \(F\) evaluated at \((\xi,t)\) and \(\xi=(u-\psi)/\eta\). Since \(e\log\chi=-m_1K\) and \(et=H-M_1-K\), differentiation gives \[eM_1=-m_1KM_1+ \chi\bigl(F_t(H-M_1-K)+F_xe\xi\bigr).\] Completing the \(H\)- and \(K\)-squares now yields \[ \begin{split} E&\ge\chi^2\mathcal C+2\chi F_xe\xi,\\ \mathcal C&= [1-l^2(1+m_1)^2]F^2 -2[1+l^2(1+m_1)]FF_t-nF_t^2. \end{split} \tag{49}\] For clarity, the discarded quantity is exactly \[|J|^2-\frac{H^2}{n-1} +\frac{(H+(n-1)\chi F_t)^2}{n-1} +\left(\frac K l-l\chi((1+m_1)F+F_t)\right)^2 +\theta\chi^2F_t^2,\] which is nonnegative. The choice of \(j\) now has two useful effects. At small slope, \(m_1\le-1/2\) leaves the coefficient of \(F^2\) at least \(3/4\), while \(\theta\chi^2=O(a^2)\) makes its eventual error small. At larger slope, the two alternatives in the pressure law supply the balance: a large value of \(F_x\) is accompanied by a favorable term \(-FF_t\).

Compact support and distance gain

Define the actual output on the original manifold by \[ h'=h+\zeta(h_u-h),\qquad t'=t+\zeta(t_u-t), \tag{54}\] extending the increments by zero. The resulting data are smooth and satisfy \(h'\ge h\) and \(t'\ge t\); their changes from \(h,t\) have compact support in \(\{r_A<W\}\). On the compact annulus \(\{.81W\le r_A\le.99W\}\), which contains every point where \(0<\zeta<1\), (42) gives \(C^3\) convergence \(u_\rho\to0\). Consequently both the uncut and cut metric and drift increments tend to zero in \(C^2\) there.

To verify the inequality after cutoff without assuming any monotonicity of \(V\), consider its residual \[\begin{split} \mathcal R(\widehat h,\widehat t)={}& \mathcal D(\widehat h,\widehat t)-\mathcal D(h,t) -(V(\widehat t)-V(t))_+\\ &+\delta(\mathop{\mathrm{tr}}_hh_0-\mathop{\mathrm{tr}}_{\widehat h}h_0)V(\widehat t) +dV(\widehat t). \end{split}\] By (53), \(\mathcal R(h_u,t_u)\ge(d/2)V(t_u)\ge dT^2/2\). The residual is continuous in the indicated \(C^2\) data, including the positive-part term. Thus \(\mathcal R(h',t')-\mathcal R(h_u,t_u)\to0\) uniformly on the annulus. For sufficiently small \(\rho\), (33) follows there. Outside this annulus, \(\zeta\) is locally constant, with value one or zero. The data then agree locally with \(h_u,t_u\), where (53) applies, or with the original \(h,t\). Completeness follows from \(h'\ge h_0\).

It remains to quantify the distance gained. Use the effective height \(\zeta u_\rho\), extended by zero. By (41) and (37), it exceeds \(W/5-1\) on \(A\) and vanishes on \(B\). Decrease \(\rho\) further so that \(|\mathrm d(\zeta u_\rho)|_h\le.01\) throughout the cutoff transition. Suppose a piecewise smooth curve from \(A\) to \(B\) had \(h'\)-length at most \(L_*W\). Its \(h\)-length is also at most \(L_*W\), and is at least \(W\). The absolute height variation in the cutoff transition, together with that on \(\{\zeta=1,s<.05\}\), is at most \(.06L_*W\); repeated visits are included by integrating the absolute derivative along the curve. Where \(\zeta=0\) there is no height variation. Thus the remaining portion, where \(\zeta=1\) and \(s\ge.05\), carries variation greater than \[(1/5-.06L_*)W-1>.11W.\] On that portion \(\mu=1\) and \(h'=h+\mathrm du_\rho^2\). Everywhere else \(h'\ge h\). Applying the integral triangle inequality to the pair consisting of the \(h\)-speed and the absolute height derivative restricted to this portion gives \[\operatorname{length}_{h'}(\gamma) >\sqrt{W^2+(.11W)^2}>L_*W,\] a contradiction. This proves (34) and completes the proof of Lemma 9.

Summing the local losses

The following form of the round argument accommodates a drift of either sign. It extends the iteration in (OpenAI 2026, sec. 4) using the positive-part term in Lemma 9.

Proof of Theorem 1

For a quadratic-form lower bound \(\lambda>0\), the metric \(\lambda g\) has normalized lower bound \(1\), since its scalar curvature and Laplacian are \(\lambda^{-1}\) times those of \(g\). Its distances are \(\sqrt\lambda\) times those of \(g\), giving the bound \(C_n/\sqrt\lambda\) stated in the introduction.