Separation between invariants of N and E_{I,ε} for some (or for all) parameter pairs

Determine whether there exists a model of ZFC in which cov(N) < cov(E_{I,ε}) for some (or for all) pairs (I, ε) with I a partition of ω into finite nonempty intervals and ε ∈ ℓ^1_+, and dually whether there exists a model in which non(E_{I,ε}) < non(N) for some (or for all) such pairs.

Background

Beyond the global separation problem, the authors ask for separations that may hold for some specific parameter pairs (I, ε), or uniformly for all such pairs. This refines the comparison between the new ideals E_{I,ε} and the classical null ideal N by focusing on existence of models with strict inequalities at the covering and uniformity coordinates.

Such results would parallel the rich forcing landscape around Cichoń’s diagram, now transplanted to the layered ideals E_{I,ε} defined via partitions and summable sequences.