A fixed cap product glues by unique transverse flow roots

Use the fixed smooth field $V$ supplied in the statement. Choose finitely many smaller foliation boxes covering $B(W)$, with compact cores and larger boxes still inside the domain of $V$. Sign their transverse coordinates so $d{z}_j(V)>0$ on the larger boxes. No foliation-coordinate field is asserted to be smooth, and $V$ is not replaced.

The transported transversals. Use the supplied positively oriented $C^2$ transversal $\tau$ at $B(x_{\ast })$ and a finite subdivision of $W$ into closed cells each mapped by $B$ into one box of the cover. Along the tree of cells, plaque continuation of $\tau$ defines for every cell $c$ a $C^2$ map $(x,t)\mapsto T_x(t)$ on $c\times J_0$, for some interval $J_0$ about $0$, with $T_x(0)=B(x)$; this uses the finite holonomy-trivial continuation hypothesis. Path independence of the germ away from the basepoint is [F5], and the regularity of each finite transport is [F4], so the assignment is $C^2$ in $(x,t)$ on each cell and the assigned pieces agree on overlaps. A finite intersection of the finitely many domains of definition gives one uniform interval $J_0=(-{\delta }_0,{\delta }_0)$ on which all these transports are defined.

In smooth ambient charts, [F2] supplies the local flow of $V$; uniqueness glues the finitely many formulas near the compact image $B(W)$. Shrink one common time interval so these flow segments stay in the relevant larger boxes. The flow is jointly $C^2$, and $s\mapsto z_j({\mathrm{\Phi }}_s(B(x)))$ is strictly increasing on each such short segment. Completeness is unnecessary.

Reduction of the root equation to one cell. Fix a cell $c$ contained in a box $U$ with transverse coordinate $z$, and put $G(s,x,t):=z({\mathrm{\Phi }}_s(B(x)))-\zeta (x,t)$, where $\zeta (x,t):=z(T_x(t))$ is jointly $C^2$ on $c\times J_0$. Since $B$ maps $c$ into one plaque of $U$, the value $z(B(x))$ is constant on $c$, and $\zeta (x,0)=z(B(x))$; since $T_x(t)$ is transported along a positive transverse direction, ${\mathrm{\partial }}_t\zeta >0$ on $c\times J_0$. Finally ${\mathrm{\partial }}_{s}G(0,x,t)=dz(V)(B(x))>0$, because $V$ is positively transverse.

At $(s,x,t)=(0,x,0)$ the root equation has value zero and ${\mathrm{\partial }}_{s}G>0$. Apply [F3] there, with $x,t$ as parameters, and cover each compact cell by finitely many of the resulting parameter neighborhoods. Shrink the common t-interval and the flow-time bound so every root remains in the short segment where ${\mathrm{\partial }}_{s}G>0$. Uniqueness then pastes these local root functions into one jointly $C^2$ function $s_c(x,t)$ on an open neighborhood of each cell times one interval $J$. Take the finite intersection of all such intervals.

On an overlap use a common smaller box around $B(x)$. The continuation data assign the same local plaque there, not just the same global leaf: transition of the transverse coordinate sends the label in one chart to the label in the other. For variable x in this overlap the central plaque label is fixed, so the equality of transverse transition germs holds on one neighborhood in x and one short interval in t; finite compact covers of the cell faces give a common interval. Both root points lie on the same short V-segment in this box and have this identical plaque label. Strict monotonicity in step 2.1 therefore gives equal flow times. Since the formulas hold on open cell neighborhoods, [F4] pastes $P(x,t)={\mathrm{\Phi }}_{s_c(x,t)}(B(x))$ jointly $C^2$ on $W\times J$.

Properties of $P$. Clearly $P(x,0)={\mathrm{\Phi }}_{s_c(x,0)}(B(x))$ and $s_c(x,0)=0$ because $G(0,x,0)=z(B(x))-\zeta (x,0)=0$ and the root is unique; hence $P(x,0)=B(x)$. For fixed $t$ all points $P(x,t)$ lie in the single leaf $L_t$ containing $\tau (t)$, since they lie in the leaf containing $T_x(t)$ and each $T_x(t)$ is obtained from $\tau (t)$ by plaque continuation; hence every slice lies in one leaf, and ${\mathrm{\partial }}_{x}P$ is tangent to $F$. For the $t$-direction, differentiating $G(s_c(x,t),x,t)=0$ gives ${\mathrm{\partial }}_t{s}_c=-{\mathrm{\partial }}_{t}G\mathrm{/}{\mathrm{\partial }}_{s}G={\mathrm{\partial }}_t\zeta \mathrm{/}{\mathrm{\partial }}_{s}G>0$, so ${\mathrm{\partial }}_{t}P={\mathrm{\partial }}_t{s}_c\cdot V({\mathrm{\Phi }}_{s_c}(B(x)))$ is a positive multiple of the positively transverse field $V$; hence every track is positively transverse and $d{P}^{-1}(TF)=TW=\ker (dt)$.

Exact collar factorization. Let $C\subseteq W$ and $f_C(x)=T_x(t_C(x))$ with $t_C(C)\subseteq J$ be as in the statement, and suppose each $f_C(x)$ is obtained by projecting along the short $V$-orbit segment of $B(x)$ used in the construction, so that $f_C(x)={\mathrm{\Phi }}_{\sigma (x)}(B(x))$ for some $\sigma (x)$ in the same short flow-time domain. Then $f_C(x)$ lies in the leaf containing $T_x(t_C(x))$, and on the $V$-orbit of $B(x)$ the equation $z({\mathrm{\Phi }}_s(B(x)))=\zeta (x,t_C(x))$ is satisfied at $s=\sigma (x)$; by uniqueness of the root in step 4.1, $\sigma (x)=s(x,t_C(x))$ and therefore $P(x,t_C(x))={\mathrm{\Phi }}_{s(x,t_C(x))}(B(x))=f_C(x)$ pointwise on $C$.

The construction used finitely many boxes, finitely many cells, finitely many bumps and finitely many local roots, so it makes only finitely many choices; the root and flow theorems of [F2] and [F3] are choice-free, and the standing hypothesis [F6] is not used beyond the pair's interface. This proves the statement.