INTERSECTION SPACES AND MULTIPLE TRANSVERSE RECURRENCE

The paper’s Theorem 1.7 states precisely the restriction-in-stages result for G = N⋊L with L preserving mN, asserting that for a finite G-invariant measure µ and a separated N-cross section Y := L·Z, N res_X^Y(µ) is finite and L-invariant, and G res_X^Z = L res_Y^Z ◦ N res_X^Y. The proof in Section 7 normalizes Haar so that mG(WNWL) = mN(WN)mL(WL) (Eq. 7.1), then uses a-injective windows V, VN, VL and the identity µ(a(C)) = (m ⊗ ν)(C) for injective C to show L-invariance and equality of transverse measures via two computations of µ(VNVL·B) (Section 7.3). This matches the candidate solution’s steps: the same Haar normalization, injectivity windows in N and L, finiteness of the intermediate transverse measure from an injective set computation, L-invariance via conjugation preserving mN, and the two-way computation leading to G res_X^Z = L res_Y^Z ◦ N res_X^Y. In short, the candidate reproduces the paper’s proof in condensed form with the same core ideas and steps. See Theorem 1.7 statement, the Haar normalization in 7.1, the injective-set machinery (Theorem 1.14/Prop. 4.3), and the restriction-in-stages proof (Theorem 7.2) for the exact correspondence .