Establish the global support-transition conjectures

Prove, for every amplitude constraint A, that new support points of the capacity-achieving input distribution for the amplitude-constrained AWGN channel appear at the origin and that the support cardinality increases by at most one at each transition.

Background

Numerical observations and prior work suggest that support transitions of the capacity-achieving input distribution are highly constrained: newly appearing points originate at zero, and the number of support points changes by at most one at a transition. The paper proves only local, unoriented versions of these claims, so their global validity remains unresolved.