  Extending our previous results, we study the double-scaling limit SYK (DSSYK)
model with an additional diagonal matrix with a fixed number $c$ of nonzero
constant entries $\theta$. This constant diagonal term can be rewritten in
terms of Majorana fermion products. Its specific formula depends on the value
of $c$. We find exact expressions for the moments of this model. More
importantly, by proposing a moment-cumulant relation, we reinterpret the effect
of introducing a constant term in the context of non-commutative probability
theory. This gives rise to a $\tilde{q}$ dependent mixture of independences
within the moment formula. The parameter $\tilde{q}$, derived from the
$q$-Ornstein-Uhlenbeck ($q$-OU) process, controls this transformation. It
interpolates between classical independence ($\tilde{q}=1$) and Boolean
independence ($\tilde{q}=0$). The underlying combinatorial structures of this
model provide the non-commutative probability connections. Additionally, we
explore the potential relation between these connections and their
gravitational path integral counterparts.
