  We develop connections between the qualitative dynamics of Hamiltonian
isotopies on a surface $\Sigma$ and their chain-level Floer theory using ideas
drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We
associate to every collection of capped $1$-periodic orbits which is `maximally
unlinked relative the Morse range' a singular foliation on $S^1 \times \Sigma$
which is positively transverse to the vector field $\partial_t \oplus X^H$ and
which is assembled in a straight-forward way from the relevant Floer moduli
spaces. We derive a purely topological and Turing-computable characterization
of the spectral invariant $c(H;[\Sigma])$ for generic Hamiltonians on arbitrary
closed surfaces. This completes, for generic Hamiltonians, a project initiated
by Humili\`{e}re-Le Roux-Seyfaddini, in addition to fulfilling a desideratum
expressed by Gambaudo-Ghys seeking a topological characterization of the
Entov-Polterovich quasi-morphism on $Ham(S^2)$.
