A partially homogeneous signed projection limit is a minimizer

Use the natural slabs $Q_r=[-r,r]\times \mathrm{\Theta }$; their cutoff and bounded supporting-selection hypotheses are verified in the box-separation proof [F5], so the initial bounded-weight premise suffices for its characteristic and support conclusions. By [F2] the limit is signwise stationary. If it were not a minimizer, [F3] would force $H=G_s$ to be negative somewhere for one sign $s\in \{+,-\}$. Fix that sign, put $m_t=\int w\phi (\theta )d{\mu }_t$, and write $H_t(\theta )=sd{R}_{m_t}[\phi (\theta )]+\tilde{V}(\theta )$ for the time-dependent risk field of the same original flow. Choose a negative regular level $a_0$ and an interior point ${\theta }_{\ast }$ of $K_0=\{H\le a_0\}$; continuity at a negative point and density of regular values allow $H({\theta }_{\ast })<a_0<0$. By [F6] choose a larger negative outer trap $K_{\mathrm{o}\mathrm{u}\mathrm{t}}$ containing $K_0$, and close-forcing radius $\rho >0$, so that $H_t<-b<0$ there. In the unbounded case choose the outer level regular for both finite and limiting sphere functions, as provided by [F6]. Reduce the forcing radius if needed to match the escape neighborhood in [F4] for the selected sign and inner set $K_0$.

By [F2], the projections eventually remain in that BL neighborhood and the forcing eventually remains within $\rho$ of $f_{\nu }$. Fix such a time $t_0$ first. Only now use [F5] to choose $M<\mathrm{\infty }$ bounding the weights of the entire support at $t_0$. Use the proof coordinate $z=sw$; the model and its feature moment $m_t$ remain unchanged. The constant-penalty conclusion in [F6] gives $\tilde{V}=c\ge 0$ and $G_{-s,t}=2c-H_t$. On a characteristic with $z<0$ and $\theta \in K_{\mathrm{o}\mathrm{u}\mathrm{t}}$, the original flow equations become $z^{\mathrm{\prime }}=2c-H_t(\theta )\ge b,{\theta }^{\mathrm{\prime }}=\mathrm{\mid }z\mathrm{\mid }\mathrm{\nabla }{H}_t(\theta )\mathrm{.}$ Consequently, when this angular path is traced backward, it satisfies the favorable equation $\dot{\theta }=-\rho \mathrm{\nabla }H$ from [F6] with the positive continuous scale $\rho =\mathrm{\mid }z\mathrm{\mid }$ and reversed but still close forcing. The universal reversed-forcing trap in [F6] keeps it in $K_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and hence keeps $z^{\mathrm{\prime }}\ge b$. Now fix $t>t_0+M\mathrm{/}b$ and suppose a support point over $K_0$ has $z(t)<0$. Trace its characteristic backward on the maximal interval ending at $t$ on which $z<0$. The reversed trap is first applied on compact subintervals where $\mathrm{\mid }z\mathrm{\mid }>0$; closedness of $K_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and continuity extend membership to a possible zero endpoint. If it starts at a time $u>t_0$ with $z(u)=0$, integrating $z^{\mathrm{\prime }}\ge b$ from $u$ to $t$ contradicts $z(t)<0$. If it reaches $t_0$, then $z(t)\ge z(t_0)+b(t-t_0)\ge -M+b(t-t_0)>0$, also a contradiction. Thus no support point over $K_0$ has $z<0$ after $t_0+M\mathrm{/}b$. By [F5], every support point has a characteristic representative, so this holds for the full support.

Fix $t>t_0+M\mathrm{/}b$. By [F5] the vertical line over ${\theta }_{\ast }$ meets $\mathrm{spt}{\mu }_t$, and by step 2.1 one such point has $z\ge 0$. If $z>0$, the strict inequality $H({\theta }_{\ast })<a_0$ puts this support point in the interior of the favorable cone, so an open neighborhood of it has positive measure. If $z=0$, write $a_t(\theta )=d{R}_{m_{{\mu }_t}}[\phi (\theta )]$. The scalar potential in the isometric coordinate $z=sw$ is $s{a}_t(\theta )z+c\mathrm{\mid }z\mathrm{\mid }$. Since $H_t({\theta }_{\ast })=s{a}_t({\theta }_{\ast })+c<-b$, its minimum-norm subgradient at $z=0$ gives $z^{\mathrm{\prime }}=-H_t({\theta }_{\ast })>b$, and the angular velocity is zero. In a small spacetime neighborhood of this point, the favorable equation on $z>0$ has $z^{\mathrm{\prime }}=-H_t>b\mathrm{/}2$, while the opposite equation on $z<0$ has $z^{\mathrm{\prime }}=2c-H_t>b\mathrm{/}2$. Continuity of the characteristic and its integral equation therefore give $z(t+h)>0$ for every sufficiently small $h>0$, while $\theta (t+h)$ remains strictly inside $K_0$. Its image is still a support point by [F5], so again there is positive favorable-cone mass. In either case [F4] forces a later exit from the neighborhood in which all projections after $t_0$ remain. This contradiction proves global optimality for the original flow, with the original feature moment unchanged.

The negative-branch contradiction in steps 1.1–3.1 uses only the existence of a negative value of $G_s$, signwise stationarity and the given flow; hence both $G_{+}$ and $G_{-}$ are nonnegative everywhere. Let ${\widehat{\mu }}_{\nu }$ be the Jordan-sign lift from [F1], supported on the sections $w=1$ and $w=-1$. It is a finite nonnegative Borel measure, has feature moment $m_{\nu }$, and satisfies $F({\widehat{\mu }}_{\nu })=J(\nu )<\mathrm{\infty }$. Its first variation at an arbitrary $(w,\theta )\in \mathbb{R}\times \mathrm{\Theta }$ is $d{R}_{m_{\nu }}[w\phi (\theta )]+\mathrm{\mid }w\mathrm{\mid }\tilde{V}(\theta )$, equal to $w{G}_{+}(\theta )$ for $w>0$, $(-w)G_{-}(\theta )$ for $w<0$, and $0$ for $w=0$. It is therefore nonnegative everywhere. By signwise stationarity [F2], it vanishes ${\widehat{\mu }}_{\nu }$-almost everywhere. Thus [F8] makes ${\widehat{\mu }}_{\nu }$ a global minimizer of $F$ over all finite nonnegative Borel measures, including finite-energy competitors without a finite $\mathrm{\mid }w\mathrm{\mid }$ moment.

By [F7], $F({\mu }_t)\to J(\nu )$, and step 3.1 gives $J(\nu )={\inf }_{\lambda }J(\lambda )$. Step 4.1 gives $J(\nu )=F({\widehat{\mu }}_{\nu })={\inf }_{\mu \ge 0}F(\mu )$ on the unrestricted finite-nonnegative-measure domain. Consequently $F({\mu }_t)\to J(\nu )={\inf }_{\lambda }J(\lambda )={\inf }_{\mu \ge 0}F(\mu )$, as asserted.