On the C8/3-Regularisation of Simultaneous Binary Collisions in the Planar 4-Body Problem

The paper proves that, for all positive masses, the SBC in the planar 4‑body problem is exactly C^{8/3}‑regularisable (Theorem 4.12) by: (i) double Levi–Civita regularisation and a Sundman time change, (ii) a blow‑up yielding a collision manifold foliated by homoclinic connections to a normally hyperbolic manifold of saddle equilibria, (iii) factoring the block map as D2 ∘ T ∘ D1, with D1,D2 hyperbolic (Dulac‑type) transitions and T a smooth transition along the homoclinic channel, and (iv) composing precise asymptotics to obtain an O(ε^{8/3}) term that obstructs C^3 while ensuring a C^{8/3} extension; the upper bound uses restriction to the collinear submanifold where exact C^{8/3} is known . The candidate solution follows essentially the same blueprint and reaches the correct conclusion, with two notable inaccuracies: it attributes the C^{8/3} regularity primarily to limited smoothness of the local Dulac maps (where the paper shows these have O(ε^3 ln ε) behavior and the critical 8/3 exponent arises from the smooth transition’s eighth‑order normal‑form term composed with the 1/3 scaling), and it describes the channel as “heteroclinic” between two components rather than the paper’s homoclinic foliation to a single NHIM .