This is the first part of an article in two parts,
which builds the foundation of a Floer-theoretic invariant, $I_{\textrm{F}}$.
The Floer homology can be trivial in many variants of the
Floer theory; it is therefore interesting to consider more refined
invariants of the Floer complex. We consider one such
instance --- the Reidemeister torsion $\tau_{\textrm{F}}$ of the Floer--Novikov complex
of (possibly non-Hamiltonian) symplectomorphisms. $\tau_{\textrm{F}}$ turns
out not to be invariant under Hamiltonian \hbox{isotopies}, but this
failure may be fixed by introducing certain "correction term'':
We define a Floer-theoretic zeta function
$\zeta_{\textrm{F}}$, by counting perturbed pseudo-holomorphic tori
in a way very similar to the genus 1 Gromov invariant. The main
result of this article states that under suitable monotonicity
conditions, the product $I_{\textrm{F}}:=\tau_{\textrm{F}}\zeta_{\textrm{F}}$ is
invariant under Hamiltonian isotopies. In fact, $I_{\textrm{F}}$ is
invariant under general symplectic isotopies
when the underlying symplectic manifold
$M$ is monotone.
Because the torsion invariant we consider is
not a homotopy invariant, the continuation method used in typical
invariance proofs of Floer theory does not apply; instead,
the detailed bifurcation analysis is worked out. This is the first time such
analysis appears in the Floer theory literature in its entirety.
Applications of $I_{\textrm{F}}$, and the construction of
$I_{\textrm{F}}$ in different versions of Floer theories are
discussed in sequels to this article [Y.-J.L.].