Prove
Lemma 41 ([Compression of a separated upper space]). Let A=A^* on a Hilbert…Lemma 42 ([An elementary contour estimate]). Suppose B=B^* has a reducing…Lemma 29 ([Equivariant trace and size]). Let L be a countable group with a…Proposition
Apply Lemma 41 ([Compression of a separated upper space]) to O(t),E_*(t),Π with a=0 and b=χ . Extend its two-part comparison operator by zero on (1-Π) H_t^0 . The extended complementary part is still at most 1-χ , because 1-χ>0 . Denote its upper projection by E_*^Π(t) . Equations, , and show that T_P(t) differs from this comparison operator by at most α for almost every t . The passage from the full regular norm in Equation to this almost-everywhere statement is the direct-integral norm formula from the fiber construction. Lemma 42 ([An elementary contour estimate]) with r_0=χ/3 gives a common contour, Equation, and \|E_P(t)-E_*^Π(t)\| ≤(α)/(r_0-α)<1. All these fields are measurable by the resolvent formulas, and their uniform bounds define the corresponding bounded direct-integral operators. The norm bound also follows directly from \|ΠOΠ\|≤1 and η<1 . Both the compression equivalence and the close-projection equivalence commute with the fiber right actions used to define the identity trace. Lemma 29 ([Equivariant trace and size]) and Lemma 37 ([The model band and its trace]) therefore imply the equality of traces in Equation, with no approximation error. Outside the window the model is zero and \|T_P(t)\|≤η almost everywhere. Consequently its contour projection is exactly zero there. This conclusion uses the small norm of the Fourier-mask leakage supplied by Theorem 9.1; it does not require the finite Fourier masks to vanish exactly. Since tr_ id p(t)>0 on the interval of positive measure (0,δ) , Equation makes E_P nonzero. Every component subspace reduces the actual adjacency T_P , and hence its resolvent and E_P . Left deck translations commute with T_P , since its entries are right deck translations with the prescribed matrix coefficients. They therefore commute with E_P . They send any lift of a fixed base vertex to every other lift, proving the last assertion, even when they permute different connected components.