Partial-homogeneity flows preserve box separation

The natural partial-model cutoffs $Q_r=[-r,r]\times \mathrm{\Theta }$ satisfy the feature-energy conditions: their distance enlargement is $Q_{r+r^{\mathrm{\prime }}}$, boundedness of $\phi ,\tilde{V}$ bounds $\mathrm{\Phi },V$ there, and the bounded Lipschitz first derivatives give Lipschitz $d\mathrm{\Phi }$ on each slab. The ambient potential has a local supporting vector $(\mathrm{sign}(w)\tilde{V},\mathrm{\mid }w\mathrm{\mid }\mathrm{\nabla }\tilde{V})$ off $w=0$, and $(0,0)$ at $w=0$; its norm is bounded by $C(1+r)$. These vectors belong to the relative supporting set by global semiconvexity, so no bound on normal vectors is needed. Thus bounded initial weights supply the common-cutoff hypothesis of [F1]. The path $t\mapsto m_{{\mu }_t}$ is continuous on $[0,T]$, so its differential image $\{d{R}_{m_{{\mu }_t}}:0\le t\le T\}$ is compact and bounded. Boundedness of $\phi ,\tilde{V}$ and [F4] give a finite $B_T$ with $\mathrm{\mid }\dot{w}\mathrm{\mid }\le B_T$ for every characteristic. Choose $\alpha >r_0+B_{T}T$. Characteristics from $K_0$ have $\mathrm{\mid }{w}_t\mathrm{\mid }<\alpha$ for $0\le t\le T$, while characteristics starting on either plane $w=\pm \beta$ remain outside the slab $\mathrm{\mid }w\mathrm{\mid }\le \alpha$ for every $\beta >\alpha +B_{T}T$.

In the bounded branch, $K_0$ is compact because it is closed and contained in the compact set $[-r_0,r_0]\times \mathrm{\Theta }$. Thus $X_t(K_0)$ is compact and closed, so [F1]–[F2] give $\mathrm{spt}{\mu }_t=X_t(K_0)$. The Neumann premise makes the angular characteristic tangent to $\mathrm{\partial }\mathrm{\Theta }$. In local boundary charts the characteristic equation restricted to $\mathbb{R}\times \mathrm{\partial }\mathrm{\Theta }$ therefore has a solution starting at every boundary point; ambient uniqueness identifies it with the characteristic from that point. This proves the forward invariance of the boundary used below.

In the unbounded branch use $\psi (\theta )=\tanh \mathrm{\mid }\theta \mathrm{\mid }\theta \mathrm{/}\mathrm{\mid }\theta \mathrm{\mid }$ for $\theta \mathrm{\ne }0$ and $\psi (0)=0$. This maps ${\mathbb{R}}^{d-1}$ diffeomorphically onto the open unit ball. Its radial and tangential differential eigenvalues are ${\mathrm{sech}}^2\mathrm{\mid }\theta \mathrm{\mid }$ and $\tanh \mathrm{\mid }\theta \mathrm{\mid }\mathrm{/}\mathrm{\mid }\theta \mathrm{\mid }$, both tending to zero at infinity. A bounded $C^1$ function with Lipschitz derivative has bounded derivative: Taylor's inequality in the unit direction of its derivative would otherwise force its bounded range to grow without bound. Apply this to scalar evaluations of $\phi$ and to $\tilde{V}$; hence the angular gradients $\mathrm{\nabla }{G}_f$ are bounded on bounded sets of $f$. For each $f$, the assumed $C^1$ limit of $G_f(r\cdot )$ gives pointwise convergence of bounded linear operators $f\mapsto ⟨f,\phi (r\cdot )⟩$ into $C^1({\mathbb{S}}^{d-2})$. On bounded radial intervals their operator norms are bounded by local $C^1$ regularity; convergence at infinity makes the family pointwise bounded. By [F5] the operator norms are uniformly bounded, and the convergence is uniform on the compact set $\{d{R}_{m_{{\mu }_t}}:0\le t\le T\}$ by finite-net approximation. The partial characteristic equations for $w>0$ and $w<0$ are $\dot{w}=-G_{a_t,+}(\theta )$, $\dot{\theta }=-w\mathrm{\nabla }{G}_{a_t,+}(\theta )$ and $\dot{w}=G_{a_t,-}(\theta )$, $\dot{\theta }=-\mathrm{\mid }w\mathrm{\mid }\mathrm{\nabla }{G}_{a_t,-}(\theta )$, respectively, where $G_{a_t,+}=a_t[\phi ]+\tilde{V}$ and $G_{a_t,-}=-a_t[\phi ]+\tilde{V}$. At $w=0$, the scalar inclusion is $\dot{w}\in -{\mathrm{\partial }}_w(a_t[\phi (\theta )]w+\tilde{V}(\theta )\mathrm{\mid }w\mathrm{\mid })$ and $\dot{\theta }=0$; the pointwise minimum-norm choice is used by the flow definition, but the vector-field selection can jump across $w=0$. The compactified angular speed is $D\psi (\theta )\dot{\theta }$, which tends uniformly to zero as $\mathrm{\mid }\theta \mathrm{\mid }\to \mathrm{\infty }$ for bounded $w$. The coefficients of the limiting scalar inclusion converge uniformly on the compact set of risk gradients. Monotonicity of the subdifferential of the convex function $w\mapsto aw+V\mathrm{\mid }w\mathrm{\mid }$ gives $\mathrm{\mid }{w}_1(t)-w_2(t)\mathrm{\mid }\le \mathrm{\mid }{w}_1(0)-w_2(0)\mathrm{\mid }+{\int }_0^t(\mathrm{\mid }{a}_1-a_2\mathrm{\mid }+\mathrm{\mid }{V}_1-V_2\mathrm{\mid })ds$ for two solutions, so the scalar characteristics converge continuously to their unique boundary limit. Write $Y_t$ for the characteristic flow in $(w,\psi (\theta ))$ coordinates; it extends continuously to the closed unit ball, with lateral angular coordinate fixed at the boundary.

In the unbounded branch the original map $X_t$ is closed on $\mathbb{R}\times {\mathbb{R}}^{d-1}$. If $X_t(w_n,{\theta }_n)$ converges to a finite point, [F4] bounds $w_n$; compactness of the closed unit ball gives a convergent subsequence of $\psi ({\theta }_n)$. Its limit cannot lie on the boundary because $Y_t$ preserves that boundary, while the image limit is finite. Hence a subsequence of $(w_n,{\theta }_n)$ converges in the original domain; closedness follows. Now [F1]–[F2] give $\mathrm{spt}{\mu }_t=X_t(K_0)$ in this branch as well.

Fix any $\beta >\alpha +B_{T}T$. In either branch, use $Y_s=X_s$ in the bounded case and the compactified characteristic $Y_s$ from step 3.1 in the unbounded case. Put ${\widehat{K}}_0^{\beta }=K_0$ in the bounded case and let ${\widehat{K}}_0^{\beta }$ be the closure of $(\mathrm{i}\mathrm{d}\times \psi )(K_0)$ in $[-\beta ,\beta ]\times \bar{D}$ in the unbounded case. This is closed; any added limit points lie on the lateral boundary because $K_0$ is closed in the original domain. Work in $U_{\beta }=(-\beta ,\beta )\times D$, where $D=\mathrm{int}\mathrm{\Theta }$ in the bounded case and $D$ is the open unit ball in the compactified case. Let $A_{\beta }$ be the component of $U_{\beta }\setminus {\widehat{K}}_0^{\beta }$ containing the connected positive face $\{\alpha \}\times D$. The positive face lies in $A_{\beta }$ because it misses ${\widehat{K}}_0^{\beta }$. No point of the negative face lies in $A_{\beta }$: a path in $A_{\beta }$ from $-\alpha$ to $+\alpha$ would have a subpath between its last crossing of $w=-r_0$ and its first later crossing of $w=r_0$, wholly inside the original box and disjoint from $K_0$, contrary to the assumed separation. Here components of an open subset of Euclidean space are path connected, since each path component is open and its complement is a union of open path components. The boundary satisfies $\mathrm{\partial }{A}_{\beta }\subseteq {\widehat{K}}_0^{\beta }\cup \mathrm{\partial }{U}_{\beta }$. At time $s\in [0,T]$, the image of the original part of ${\widehat{K}}_0^{\beta }$ lies in $\mathrm{\mid }w\mathrm{\mid }<\alpha$, the horizontal boundary planes $w=\pm \beta$ map outside $\mathrm{\mid }w\mathrm{\mid }\le \alpha$, and the lateral boundary maps into itself by steps 2.1–3.1. The added compactification points are on that lateral boundary. Therefore every target on either $\alpha$ face avoids $Y_s(\mathrm{\partial }{A}_{\beta })$. The definition and properties of $D$ in [F3] give $D(Y_t,A_{\beta },y)=1$ on the positive face and $D(Y_t,A_{\beta },y)=0$ on the negative face.

Suppose a path in $\mathbb{R}\times \mathrm{\Theta }$ joins the negative to the positive $\alpha$ face without meeting $X_t(K_0)$. In the bounded branch, compactness and positive distance from the closed support let us perturb the path by an arbitrarily small convex interpolation toward a fixed interior point of $\mathrm{\Theta }$; the perturbed path stays disjoint from the support and has endpoints in $D$. In the unbounded branch its compact image already lies in the open ball after applying $\psi$. Stop the path at its first visit to the positive face, and let $m$ be its minimum $w$-coordinate. Choose $\beta >\max \{\alpha ,-m\}+B_{T}T$ and use the construction of step 5.1 with this $\beta$. The target path is contained in $U_{\beta }$ and avoids the images of the horizontal boundary planes. In the unbounded case it lies in the open ball, so it also avoids the added lateral limit points of $Y_t({\widehat{K}}_0^{\beta })$; its remaining points avoid that image by the assumption that the original path misses $X_t(K_0)$. Degree invariance along this target path would make its endpoint values of $D(Y_t,A_{\beta },\cdot )$ equal, contradicting the values $0$ and $1$ from step 5.1. Thus every such path intersects $X_t(K_0)=\mathrm{spt}{\mu }_t$.