This exact 12-point set is non-extendable: no 13th point in R^4 preserves the almost-equidistant property. Criterion (proved): adding y stays almost-equidistant iff the set of non-unit-neighbours of y is a clique of the unit-distance graph; equivalently y is at unit distance from every vertex outside some clique T. Testing the 8 maximal cliques (six K5, two K3), each linear system {y in aff(P); |y-p|^2=1 for all p outside T} is infeasible over R.