  In this paper, we study the probabilistic local well-posedness of the cubic
Schr\"odinger equation (cubic NLS): \[ (i\partial_{t} + \Delta) u = \pm |u|^{2}
u \text{ on } [0,T) \times \mathbb{R}^{d}, \] with initial data being a Wiener
randomization at unit scale of a given function $f$. We prove that a solution
exists almost-surely locally in time provided \(f\in
H^{S}_{x}(\mathbb{R}^{d})\) with
\(S>\max\big(\frac{d-3}{4},\frac{d-4}{2}\big)\) for \(d\geq 3\). In particular,
we establish that the local well-posedness holds for any \(S>0\) when \(d=3\).
We also show that, under appropriate smallness conditions for the initial data,
the solutions are global in time and scatter.
  The solutions are constructed as a sum of an explicit multilinear expansion
of the flow in terms of the random initial data and of an additional smoother
remainder term with deterministically subcritical regularity. This construction
allows us to introduce a new and refined notion of graded scattering.
  We develop the framework of directional space-time norms to control the
(probabilistic) multilinear expansion and the (deterministic) remainder term
and to obtain improved bilinear probabilistic-deterministic Strichartz
estimates.
