3− 2− 1 FOLIATIONS FOR REEB FLOWS ON THE TIGHT 3-SPHERE

The paper proves that under hypotheses (i)–(v) there exists a 3–2–1 finite‑energy foliation on S^3 with bindings P1, P2, P3 and, as a corollary, a homoclinic orbit to P2. It builds a 1–parameter family of embedded planes asymptotic to P3 (Theorem 3.3), analyzes its ends via SFT compactness to produce two cylinders from P3 to P2 and a plane at P2 (Proposition 3.5), assembles the torus T := V1 ∪ V2 ∪ P2 ∪ P3 dividing S^3 into R1 and R2, then constructs a cylinder U from P2 to P1 using a 1–dimensional moduli of generalized planes at P2 (Section 4), and finally glues to obtain a family of cylinders from P3 to P1 and the full foliation; the existence of a homoclinic to P2 is forced by a careful transversality/area argument on a small torus around P2 (end of §5 and §6). These steps and the definition of a 3–2–1 foliation (including the pieces V1, V2, D, U and families F_τ, C_τ) appear explicitly in the paper’s statement and proofs (Definition 1.4; Theorem 1.5 and its proof; Theorem 3.3; Proposition 3.5; Theorem 5.1; Proposition 5.3; conclusion of §5–§6) . The candidate solution follows the same skeleton: it constructs the P3‑plane moduli, identifies the two breaking cylinders to P2 and the P2‑plane, builds T and regions R1,R2, then finds the P2→P1 cylinder and glues to obtain the P3→P1 family; it also asserts that all regular leaves are strong sections, consistent with the paper’s discussion (Remark 1.3) . The only substantive divergence is in the final step: the candidate argues for a homoclinic via stable/unstable “slope” intersection on a small torus, whereas the paper supplies a detailed transversality and area‑type argument (Lemma 6.5, Proposition 6.4, and the end of §5) to force the homoclinic . As written, the candidate’s homoclinic step is a plausible outline but lacks the paper’s rigorous justification. Overall, both arrive at the same conclusion by substantially the same route; the paper fills in key technical points.