Nonoptimal partial projections have an escaping sign branch

Boundedness and Lipschitz differential of $\phi$ give a finite $C_{\phi }=\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }D\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$ on either domain alternative. In the unbounded alternative, if $L=\mathrm{Lip}(D\phi )$, Taylor's inequality on a unit segment gives $\mathrm{\parallel }D{\phi }_{\theta }e\mathrm{\parallel }\le 2\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}+L\mathrm{/}2$ for every unit vector $e$; the same argument bounds $D\tilde{V}$. In the bounded alternative the derivative bounds follow from compactness. For any finite signed $\sigma$, test its BL norm with $⟨z,\phi ⟩\mathrm{/}{C}_{\phi }$ for unit $z$ to obtain $\mathrm{\parallel }\int \phi d\sigma \mathrm{\parallel }\le C_{\phi }\mathrm{\parallel }\sigma {\mathrm{\parallel }}_{BL}$ (if $C_{\phi }=0$ the assertion is immediate). Local Lipschitz continuity of $\mathrm{\nabla }R$ near $m_{\nu }$ then bounds $\mathrm{\parallel }{f}_t-f_{\nu }\mathrm{\parallel }$ by $C\mathrm{\parallel }{h}^1({\mu }_t)-\nu {\mathrm{\parallel }}_{BL}$ whenever this BL distance is small. This is a norm bound on the Hilbert forcing, on bounded and unbounded $\mathrm{\Theta }$ alike.

By [F1] write $\tilde{V}=c$. Apply [F2] to each nonempty $K_s$ using its level $-{\eta }_s$ as the inner threshold; if necessary choose a regular outer level strictly between that threshold and zero, also regular for the sphere limit in the unbounded case. The common forcing radius $\rho >0$ and constant $b>0$ can be chosen for the at most two branches. Whenever $\mathrm{\parallel }{f}_t-f_{\nu }\mathrm{\parallel }<\rho$, every favorable characteristic starting with $z=sw>0$ and $\theta \in K_s$ stays in the corresponding outer negative trap, satisfies $z^{\mathrm{\prime }}=-G_{s,t}(\theta )\ge b$, and retains its sign. Indeed its angular equation is ${\theta }^{\mathrm{\prime }}=-z\mathrm{\nabla }{G}_{s,t}$, and time change by the positive factor $z\mathrm{/}k(\theta )$ gives precisely the normalized curve in [F2]. On each finite interval all quantities are finite and $z\ge z(t_0)>0$, so this time change is legitimate; if its total normalized time is finite, the same trap follows on every finite physical interval.

Choose $ϵ>0$ so that step 1.1 sends BL distance less than $ϵ$ into the forcing ball in step 2.1. Suppose, to the contrary, that after $t_0$ this distance always remains less than $ϵ$. By hypothesis the set of superposed paths starting in the union of the favorable cones has probability $P={\mu }_{t_0}(A)>0$. Step 2.1 gives $\mathrm{\mid }{w}^{\mathrm{\prime }}(t)\mathrm{\mid }\ge b$ along every such path for almost every later time. Thus $A(t)\ge P{b}^2$ for almost every $t>t_0$. Integrating on the infinite time interval contradicts the finite total action in [F3]. Hence there is a finite $t_1>t_0$ with distance at least $ϵ$, as claimed. This action lower bound is positive before any signed projection is taken, so no opposite-sign mass can cancel it.