Attraction-Repulsion Swarming: A Generalized Framework of t-SNE via Force Normalization and Tunable Interactions

The paper states and proves the two-sided diameter differential inequality for ARS in Proposition 4.1 using a projection argument on arbitrary unit vectors, yielding equation (4.2) from the dynamics (4.1) with the collective coefficients αij(t) and the ARS flow (2.9a) . The candidate solution proves the same bound via a diameter-attaining pair, Dini derivatives, and an explicit summation that evaluates to −N|yi−yj|, thereby producing both sides of (4.2). The two arguments are logically consistent and reach the same conclusion; the model’s proof is a different, slightly more detailed route to the same inequality.