  Based on the contact Hamiltonian Floer theory established by Will J. Merry
and the second author that applies to any admissible contact Hamiltonian system
$(M, \xi = \ker \alpha, h)$, where $h$ is a contact Hamiltonian function on a
Liouville fillable contact manifold $(M, \xi = \ker \alpha)$, we associate a
persistence module to $(M, \xi, h)$, called a gapped module, that is
parametrized only by a partially ordered set. It enables us to define various
numerical Floer-theoretic invariants. In this paper, we focus on the contact
spectral invariants and their applications. Several key properties are proved,
which include stability with respect to the Shelukhin-Hofer norm in contact
geometry and a triangle inequality of contact spectral invariants. In
particular, our stability property does not involve any conformal factors; our
triangle inequality is derived from a novel analysis on pair-of-pants in the
contact Hamiltonian Floer homology. While this paper was nearing completion,
the authors were made aware of upcoming work by Dylan Cant, where a similar
persistence module for contact Hamiltonian dynamics was constructed.
