  In this paper, we present a comprehensive toolbox for studying Carrollian
stretched horizons, encompassing their geometry, dynamics, symplectic geometry,
symmetries, and corresponding Noether charges. We introduce a precise
definition of ruled stretched Carrollian structures (sCarrollian structures) on
any surface, generalizing the conventional Carrollian structures of null
surfaces, along with the notions of sCarrollian connection and sCarrollian
stress tensor. Our approach unifies the sCarrollian (intrinsic) and stretched
horizon (embedding) perspectives, providing a universal framework for any
causal surface, whether timelike or null. We express the Einstein equations in
sCarrollian variables and discuss the phase space symplectic structure of the
sCarrollian geometry. Through Noether's theorem, we derive the Einstein
equation and canonical charge and compute the evolution of the canonical charge
along the transverse (radial) direction. The latter can be interpreted as a
spin-2 symmetry charge.
  Our framework establishes a novel link between gravity on stretched horizons
and Carrollian fluid dynamics and unifies various causal surfaces studied in
the literature, including non-expanding and isolated horizons. We expect this
work to provide insights into the hydrodynamical description of black holes and
the quantization of null surfaces.
