Proposition: Indistinguishable-trial symmetry obstruction

Suppose infinitely many indistinguishable experiment blocks may be permuted arbitrarily while preserving the same datum. No injective finite-dimensional affine representation of all their events can be covariant under every such permutation.

Proof. Choose finitely many represented points forming an affine basis of the representation's affine hull. They occupy finitely many blocks. Transpose two indistinguishable blocks outside them. Its implementing affine map fixes the basis, hence is the identity, but must exchange distinct represented events, a contradiction.