Necessity of the affine encoder characterization for three transmitters

Establish whether the affine encoder form is necessary for exact identifiable computation of functions of two aggregate statistics when there are exactly three transmitters, rather than merely sufficient.

Background

Theorem 1 proves that, for at least four transmitters and under the stated regularity and nonaffinity assumptions on the receiver postprocessing functions, every continuous encoder collection enabling identifiable computation must have the common affine form E_k(s)=c_k+u\phi_{1,k}(s)+v\phi_{2,k}(s), with u and v linearly independent over the reals. The proof relies essentially on a fourth-node comparison.

The paper observes that the affine form remains sufficient when K=3, but does not prove that all continuous encoders permitting exact computation must have this form. Determining necessity for three transmitters would therefore extend the rigidity theorem to the smallest network size not covered by the current argument.