Lemma 10 ([Compression, changes of frame, and smooth functions]). The map, for the frames of Lemma 2 ([Uniform frame localization]), preserves rapid locality. An operator supported in B_r(x) remains localized about that ball, with …

Put two slot spaces side by side. If S,T are their analysis isometries, the partial isometry TS^* has rapidly decaying matrix entries by Lemma 2 ([Uniform frame localization]). On their direct sum, the anti-self-adjoint one-particle operator 0, -ST^*; TS^*, 0 rotates the two physical subspaces into each other through angle π/2 and acts as zero on their complements. Its second quantization is a bounded-strength, rapidly local quadratic interaction: absolute row sums and every distance moment are bounded by. Lemma 7 ([Propagation of local observables]) applies over this fixed angle. Conjugate A⊗I by this rotation and take the vacuum expectation in the entire first slot copy, retaining the entire second copy. On a CAR monomial, the physical components have moved from S U_q to T U_q , while the complementary components in the first copy are evaluated in their vacuum. The resulting map is exactly, including for odd monomials; linearity gives the identity for every operator. Vacuum expectation is contractive and preserves support in the doubled slot geometry. This proves the assertion, and S=T gives re-embedded physical compression. In particular no projection onto the output complementary vacuum is left in the resulting operator. For the functional-calculus assertion extend f smoothly to a compactly supported function on ℝ . If A_r approximates A locally, its Hermitian part does so with the same error. The Fourier formula and Duhamel's identity give \|f(A)-f(A_r)\| ≤(∫_ ℝ|t|\,|f(t)|\,dt) \|A-A_r\|. Since f(A_r) belongs to the same local algebra, the approximation bounds follow. A constant value of f contributes only a scalar identity, which is local in every ball.