  We show that on any compact K\"ahler surface existence of solutions to the
Z-critical equation can be characterized using a finite number of effective
conditions, where the number of conditions is bounded above by the Picard
number of the surface.This leads to a first PDE analogue of the locally finite
wall-chamber decomposition in Bridgeland stability.
  As an application we characterize optimally destabilizing curves for
Donaldson's J-equation and the deformed Hermitian Yang-Mills equation, prove a
non-existence result for optimally destabilizing test configurations for
uniform J-stability, and remark on improvements to convergence results for
certain geometric flows.
