UNIQUENESS OF HYPERBOLIC BUSEMANN FUNCTIONS IN THE NEWTONIAN N-BODY PROBLEM

The paper proves Theorem 1.1 (uniqueness in S_a up to an additive constant) via the cone theorem (Theorem 1.2), uniqueness of geodesic rays in a truncated cone, and the differentiability/gradient identification along calibrating curves; it then extends equality to all of E^N using the calibrating identity and domination inequality (see Definition 2 and Theorem 1.1; proof completes by the global extension argument) . The candidate’s solution instead reduces uniqueness in S_a to the uniqueness of the hyperbolic Busemann (directed horofunction) for a given limit shape a and uses a Busemann-type representation along backward-calibrating curves. This is a different route: it relies on the monotonic Busemann family u_{γ,T}(x)=φ_h(x,γ(T))−φ_h(γ(0),γ(T)) along a geodesic ray and the weak KAM representation along calibrating curves (monotonicity is standard for geodesic metrics; cf. the Busemann monotonicity construction) , plus the previously established weak KAM toolkit in the N-body setting. The only substantive gap in the candidate’s write-up is that Step 2 asserts the limit exists without explicitly invoking the standard monotonicity argument; this is easily patched. A second subtlety is that the candidate cites the 2024 uniqueness-of-Busemanns result—precisely the present paper—to conclude uniqueness in S_a, whereas the paper itself proves uniqueness in S_a first and then deduces uniqueness of Busemann functions as corollaries (Corollaries 1.4–1.6) . Net: the paper’s proof is complete and correct; the model’s approach is also correct in substance (once the monotonicity step is included), but it depends on the very paper’s main theorem for the Busemann uniqueness.