Decomposition of local cohomology tables of modules with large E-depth

We introduce the notion of E-depth graded modules over polynomial rings to measure depth certain Ext modules. First, we characterize with (sufficiently) large as those whose partial general initial submodules preserve Hilbert function local cohomology supported at irrelevant maximal ideal, extending a result Herzog and Sbarra on sequentially Cohen-Macaulay Second, describe cone tables sufficiently high E-depth, building previous work second author Smirnov. Finally, obtain non-Artinian version socle-lemma proved by Kustin Ulrich.