  The 2016 papers of J. Solomon and S. Tukachinsky use bounding chains in
Fukaya's $A_{\infty}$-algebras to define numerical disk counts relative to a
Lagrangian under certain regularity assumptions on the moduli spaces of disks.
We present a (self-contained) direct geometric analogue of their construction
under weaker topological assumptions, extend it over arbitrary rings in the
process, and sketch an extension without any assumptions over rings containing
the rationals. This implements the intuitive suggestion represented by their
drawing and P. Georgieva's perspective. We also note a curious relation for the
standard Gromov-Witten invariants readily deducible from their work. In a
sequel, we use the geometric perspective of this paper to relate
Solomon-Tukachinsky's invariants to Welschinger's open invariants of symplectic
sixfolds, confirming their belief and G. Tian's related expectation concerning
K. Fukaya's earlier construction.
