Proposition: Indistinguishable-block covariance obstruction

Suppose a datum contains infinitely many identical labelled experiment blocks, and the declared matched equivalences include all finite permutations of those blocks.

There is no injective embedding of its unchanged component carrier into a finite-dimensional affine space satisfying (eq:affine-covariance) for every such self-equivalence.

In particular, no constant process family can belong to a domain satisfying both (K2) and this matched version of (K4).

Proof. Choose the corresponding component containing party $A$ in each identical block, and denote these distinct components by $p_1,p_2,\dots$. Let $v_i=\iota (p_i)$. Every transposition of two selected blocks is a matched self-equivalence, so its affine map transposes the corresponding $v_i,v_j$ and fixes the other selected points.

If the affine space has dimension $m$, choose $m+2$ of these points. There is an affine dependence

with not all coefficients zero. Apply the affine map implementing the transposition $i,j$. Because the coefficients sum to zero, its translation term cancels. Subtract the original dependence to obtain

Injectivity gives $v_i\ne v_j$, hence $a_i=a_j$. This holds for every pair, so all coefficients are equal. Their sum is zero, so all are zero, a contradiction.