Exact Stack-Alphabet Costs and Nonmonotone Saturation Gaps in Guarded Two-Layer Graph Actions

The paper advances our investigation of physical realization and assembly defects by studying guarded two-layer graph actions implemented by deterministic, one-control-state stack machines. For every root-containing upper-layer mask over a finite two-vertex-connected graph, it determines the exact minimum stack-alphabet size and constructs an optimal realization requiring stack height at most two. It then characterizes, at every alphabet budget, both the total population attainable across individually realizable masks and the largest population realizable jointly in a single mask. Their difference defines a saturation gap that can increase when the resource budget is enlarged, before eventually decreasing to zero. At the first feasible budget, this gap equals the vertex-cover number of an induced graph, yielding an NP-complete family-profile decision problem; the hardness also persists at the next budget. A further result establishes a sharp computational distinction between stack heights one and two: the former has an explicit linear saturation profile, while the latter activates vertex-cover complexity. Its main novelty is the exact connection between stack-alphabet resources, joint realizability, nonmonotone saturation gaps, and NP-completeness within a concrete guarded two-layer action family, without extending the hardness claim to arbitrary pushdown realizability or a fixed-source assembly defect.