  The high sensitivity of the spectrum and wavefunctions to boundary
conditions, termed the non-Hermitian skin effect, represents a fundamental
aspect of non-Hermitian systems. While it endows non-Hermitian systems with
unprecedented physical properties, it presents notable obstacles in grasping
universal properties that are robust against microscopic details and boundary
conditions. In this Letter, we introduce a pivotal theorem: in the
thermodynamic limit, for any non-Hermitian systems with finite-range
interactions, all spectral moments are invariant quantities, independent of
boundary conditions, posing strong constraints on the spectrum. Utilizing this
invariance, we propose a new criterion for bulk dynamical phases based on
experimentally observable features and applicable to any dimensions and any
boundary conditions. Based on this criterion, we define the bulk
dispersive-to-proliferative phase transition, which is distinct from the
real-to-complex spectral transition and contrary to traditional expectations.
We verify these findings in 1D and 2D lattice models.
