Unnumbered main result: WPICC parametrisation (8), (A5)-(A6)

Let $s,t$ be any Bell scenarios with finite nonempty question and answer sets, and let $w$ map real source arrays to real target arrays. Call a source array nonsignaling when it is nonnegative, sums to one over answers at every question pair, and each party’s answer marginal is independent of the other party’s question. Assume that, on nonsignaling arrays, $w$ is a discrete probability mixture, over an arbitrary hidden-variable set, of the following deterministic preparations: either each source question is a function of the corresponding target question and each target answer is a function of the corresponding target question and source answer; or Alice first uses a fixed source question, and her source answer chooses a continuation in which Bob either waits to choose his source question as a function of his target question, with Alice’s target answer depending only on her target question and Bob’s on his target question and source answer, or uses a fixed source question before the target questions, with both target answers depending on Bob’s source answer and the respective target question; or the same construction with Alice and Bob interchanged. Continuation choices and their functions may depend on the first source answer, and each preparation’s output probabilities are obtained by summing the source probabilities over source answers consistent with these deterministic outputs. Then there exist a real $\lambda\in[0,1]$, a map $v$, and a real-linear map $L$ on the full real array spaces such that: $v$, on every real source array, is a transformation formed by independently averaging product input kernels and product output kernels, namely

where both expectations use discrete probability mass functions on arbitrary sets and all kernels are nonnegative and normalized; for every nonsignaling $r$, $Lr$ is a nonnegative normalized array admitting a local representation $(Lr)(\chi,\psi,\alpha,\beta)=\sum_z\nu(z)A_z(\alpha\mid\chi)B_z(\beta\mid\psi)$ with a discrete probability mass function and normalized nonnegative responses; and, for every nonsignaling $r$, $wr=\lambda Lr+(1-\lambda)vr$ pointwise. The values $\lambda=0,1$ are allowed, and the last two requirements concern only nonsignaling inputs.