Proposition: Repeated-block covariance obstruction

Suppose infinitely many indistinguishable experiment blocks can be permuted arbitrarily while preserving the datum. There is no injective finite-dimensional representation that is covariant under all those permutations by affine maps.

Proof. Choose finitely many represented points that affinely span the representation's affine hull. Only finitely many blocks contain them. Transpose two indistinguishable blocks outside this finite set.

Its implementing affine map fixes every point of the chosen affine basis, hence is the identity on the affine hull. It must also exchange two distinct represented events, contradicting injectivity.