Explicit Witnesses at Every Gap of the Depth Filtration of $β\mathbb{N}$

Let $Σ_{1} = \mathbb{N}^*$ and $Σ_{k+1} = \overline{\mathbb{N}^* + Σ_{k}}$ be the cumulative depth filtration of $β\mathbb{N}$, the analogue for $(\mathbb{N},+)$ of a chain of closed ideals that Protasov and Protasova studied for discrete groups, where strict descent follows from a theorem of Lutsen