An Allocation-Invariant Diffuse Measure That Is Not Mass-Stationary

Invariance under every measurable equivariant measure-preserving allocation does not characterize mass-stationarity of a diffuse random measure without an independent stationary background. For every dimension d >= 2, a periodic measure supported on parallel hyperplanes with distinct positive weights gives an explicit obstruction. Horizontal translation symmetry prevents deterministic equivariant allocations from splitting layers, and a finite weighted quotient forces the rooted configurations to be fixed. Every probability law on the finitely many root types is allocation-invariant, whereas exactly the weight-proportional law is mass-stationary. The latter is the normalized Palm law of a stationary ergodic uniform-phase probability with finite intensity. An explicit preserving Markov kernel detects all other root laws, so the separate Markovian-kernel question is not claimed solved. Related atomic, product and invariant-direction methods are credited. This is a self-audited, unrefereed note without a priority claim.