An extension can be constructed without renaming it

The new symbol ${{\mathcal{E}}_{0,\mathrm{res}}}^{\mathrm{R}}$ is essential. Its existence is now proved, so existence of a restricted solution-operator extension is not left open. However, the construction is not a declaration that every inherited occurrence of ${E_0}^{\mathrm{R}}$ already denotes this extension.

The restricted domain contains the polarized witness forcing. It need not contain every forcing arising before polarization from real cosine and sine inputs and their sums. That larger family requires its own domain statement if family-level identification is claimed.