Determine the direction of support-cardinality transitions

Determine whether the support cardinality of the capacity-achieving input distribution for the amplitude-constrained AWGN channel is nondecreasing as the amplitude constraint A increases, and establish the direction of each local support transition.

Background

The paper establishes that, near any fixed amplitude, the support cardinality can remain unchanged or increase by one, and that nonzero support points persist locally. However, these results do not determine whether a transition occurs as A increases or decreases, nor whether support points are globally monotone in A. In particular, proving monotonicity of the support cardinality requires resolving the orientation of the local birth, death, splitting, and merging events.