Proposition 9 ([A covariant line model]). There is a model λ:C₀()⊗P⟶D for Lebesgue measure and F . Let α(f)(t)=f(t-1) , acting trivially on P . For c.p.c. coordinate lifts λ_n , there are unitaries u_n∈M(Q) such that …

For j≥2 , put I_j=(-j,j] and give it Lebesgue measure. This is a finite full-support cone measure, so Theorem 7.1 ([Existence of cone models]), transported along a homeomorphism (0,1]≅I_j , gives a model λ^ (j) :C_0(I_j)⊗P→D . Choose 0≤η_j≤1 in C_c((-j,j)) , equal to one on [-j+1,j-1] . The maps R_j:C_0()⊗P⟶C_0(I_j)⊗P, R_j(s)=η_j\,s|_ I_j , are c.p.c.: restriction followed by multiplication by the scalar positive contraction η_j is completely positive and contractive, and its values vanish at the omitted left endpoint. The domain C_0(I_j)⊗P of each λ^ (j) is separable and nuclear. The maps R_j are exactly multiplicative on every fixed finite set of compactly supported inputs once j is large enough to contain their supports in the region where η_j=1 . Take c.p.c. coordinate lifts of each λ^ (j) , by , and compose with R_j . Apply Lemma 6 ([Local diagonalization]) to an increasing dense list of compactly supported algebra tests and the elementary rank tests of Lemma 3 ([Elementary determination]). Here the tests are fixed on the line before increasing j : each function, its positive trims, and its enlarged compact support all lie inside (-j+1,j-1) in every sufficiently late row. Their Lebesgue masses are therefore unchanged by R_j . For every sandwich test choose the strict scalar slack, the enlarged source, and the trimmed target before selecting coordinates in that row. The constant Cuntz representatives in Q are the same for all these rows. The fixed target tolerances give the bounded comparison witnesses of Lemma 4 ([Bounded witnesses into a fixed cut]); thus the row diagonal retains the comparisons in D , even when the row thresholds vary. Choose the coordinates for the first j tests only after these choices, with algebra errors at most j^ -1 . The resulting c.p.c. sequence λ_n is asymptotically multiplicative. Its quotient λ satisfies the elementary rank identities for every regularized limit, and Lemma 3 ([Elementary determination]) proves the asserted model identity. Lebesgue measure is translation invariant, including the extended product convention. Hence λ and λα are models for the same data. Apply Theorem 5.1 ([Norm uniqueness]) to their specified coordinate lifts to obtain.