Corollary 4 (19): WPICC is not contained in GW

There exist scenarios $s,t$ and an operation $w$ between their real behavior arrays with the following two properties. First, on all nonsignaling source correlations, $w$ is a probability mixture, with a probability mass function on an arbitrary type, of deterministic preparations: either choose each source input from its own target input and each final output from its own target input and source outcome; or fix Alice’s input first and, for each Alice outcome, either choose Bob’s input from his target input while Alice’s output ignores Bob’s outcome, or fix Bob’s input and permit both outputs to depend on Bob’s outcome; or use the symmetric Bob-first construction. In these cases each final output can depend on its own target input and the stated available outcomes, and the resulting array sums the selected source entries with indicators enforcing those outputs. Second, there are no nonnegative normalized kernels $I(x,y\mid\chi,\psi)$ and $O(\alpha,\beta\mid a,b,x,y,\chi,\psi)$ such that, for every nonsignaling correlation $p$,

Here all four sizes of each scenario are positive integers, and a nonsignaling correlation is a nonnegative array normalized over outputs at each input pair, with Bob’s marginal independent of Alice’s input and Alice’s marginal independent of Bob’s input. The claim concerns agreement on this set; it does not require any particular behavior of $w$ elsewhere.