Example. A nonstable retained set need not inherit the action [fdip-003L]

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In the alternating four-cycle with weights \((1,t,1,t)\) and \(t,m>0\), consider the half-turn and the adjacent retained subset \(R=\{0,1\}\). The half-turn preserves the full state, but sends \(R\) to \(D=\{2,3\}\). No map \(T\) of retained fields can satisfy \(\pi _RP=T\pi _R\): the zero field forces \(T(0)=0\), while \(\delta _2\) has retained projection zero and projects to \((1,0)\) after the half-turn. The action transports the marginal to a different subset; it does not descend through this fixed projection. This does not exclude accidental symmetries of an individual two-site density. A general restricted action may also be trivial, unlike the opposite-pair swap. Extra mass fields or reference pairings would require their own preservation hypotheses.

The stable-subset requirement is the equivariance convention of [milne2025group, p. 58]. The two-field witness proves failure of factorization even through a nonlinear function. It differs from the fixed-label relabeling failure: here the full half-turn symmetry is unchanged, and the chosen projection fails to admit its action. The moving square of Proposition [fdip-003E] remains valid on \(\mathbb R^{pR}\). Over any nonzero commutative unital coefficient ring \(k\), the same delta-field witness works on \(k^V\); over the zero ring all fields coincide.