Given any pair $(U,\Gamma_{\mathrm{nor}})$ of subsets of $\mathbb{R}^{n}$, diffeomorphic to $(D^{n},D^{n-1})$ and with $\Gamma_{\text{nor}}=U\cap\{b_{1}=0\}$, a sequence $\ell=\{\ell_{k}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{\text{nor}}}$ and integers $m_{2},\ldots,m_{n}$ such that

$$\int_{[db_{j}]}\ell_{1}=m_{j},\quad\text{for all}\ j=2,\ldots,n,$$

(56)

there exists a smooth symplectic manifold $(X,\omega)$ and a stitched Lagrangian fibration $f:X\rightarrow U$ satisfying the following properties:

• (i)

  the coordinates $(b_{1},\ldots,b_{n})$ on $U$ are action coordinates of $f$ with $\mu=f^{\ast}b_{1}$ the moment map of the $S^{1}$ action;

• (ii)

  the periods $\{db_{1},\ldots,db_{n}\}$, restricted to $U^{\pm}$ correspond to bases $\gamma^{\pm}=\{\gamma_{1},\gamma_{2}^{\pm},\ldots,\gamma_{n}^{\pm}\}$ of $H_{1}(X,\mathbb{Z})$ satisfying conditions (a) and (b) prior to Proposition 6.5;

• (iii)

  there is a Lagrangian section $\sigma$ of $f$, such that $(\bar{Z}_{\text{nor}},\ell)$ are the invariants of $(X,f,U,\sigma,\gamma^{+})$.