Analytic reparametrizations of translation toral flows with countable Lebesgue spectrum

The paper proves exactly what the solver claims: an analytic time-change of a minimal translation flow on T^5 with Lebesgue spectrum of countable (hence infinite) multiplicity, and the time-change Φ can be chosen strictly positive, real entire, and arbitrarily close to 1 on any bounded domain. Theorem 1 states this explicitly (Lebesgue spectrum with infinite multiplicity; Φ real entire and arbitrarily close to 1) . The construction proceeds via a special flow over a T^4 translation with a carefully designed real-analytic “roof” ϕ (Section 3.2), and α in a set Y that guarantees at each time there are three independent directions of uniform stretch (Lemma 2) . They then prove square-summable decay for smooth coboundaries (Theorem 3) and an abstract multiplicity criterion (Theorem 4 / Theorem 6) tailored from earlier work of Fayad–Forni–Kanigowski, yielding countable Lebesgue spectrum for the special flow (Theorem 2) . Finally, they realize the special flow as a time-change of the linear flow (α,1) on T^5 using an explicit Fourier-analytic correspondence (Corollary 1), exactly as the model describes, with the small Fourier tail giving proximity to 1 on bounded domains . Minor notational conventions differ (the paper phrases the reparametrization as “by 1/Φ” in Corollary 1, whereas Theorem 1 and the model take it “by Φ”), but this is immaterial up to inversion of the speed function. The only mismatch is that the model says the roof ϕ can be taken “real entire,” whereas the paper constructs ϕ as real analytic on T^4; the “real entire” requirement is asserted for Φ on T^5 (Theorem 1), not necessarily for ϕ. Aside from this, the solution follows the paper’s method closely and correctly.