The local Floer cohomology of indicator functions

For a compact set $K$ with contact type boundary in a symplectic manifold $M$ we construct a spectral sequence from the local Floer homology of the Reeb orbits, as studied by \cite{Mclean2012}, to the relative symplectic cohomology of $K$ in $M$ over the Novikov ring. The spectral sequence is functorial with respect to inclusions which are not required to be exact. This functoriality is key to the closed string reconstruction problem near the singularity of an SYZ fibration. We illustrate this in the case of dimension $2n=4$ for symplectic cluster manifolds. In higher dimension, an additional ingredient, the locality spectral sequence, is required, and is the subject of a forthcoming work in progress.