COLLECTIVE DYNAMICS OF LOHE TYPE AGGREGATION MODELS

The paper states and sketches proofs for the Lohe tensor (LT) model’s aggregation in both homogeneous and heterogeneous settings, built on a diameter differential inequality of the form |d/dt D(T) + κ0 D(T)| ≤ 2κ0 D(T)^2 + 2κ̂0‖T_c^0‖_F D(T) + D(A) (their eq. (3.1)) and derives: (i) two-sided exponential decay in the homogeneous case under κ̂0 < κ0/(2‖T_c^0‖_F) and small initial diameter, and (ii) practical aggregation in the heterogeneous case with limsup bounds expressed via the largest root of −2κ0x^2 + (κ0 − 2κ̂0‖T_c^0‖_F)x = D(A) (Theorem 3.3). These match the candidate solution’s conclusions and proof strategy (pairwise estimates → diameter → logistic comparison, plus co-rotating frame for D(A)=0). The only discrepancy is a constant: the candidate’s intermediate pairwise-to-diameter inequality uses a + (κ0/2) D^2 term, whereas the paper’s key inequality (3.1) carries + 2κ0 D^2. However, the candidate ultimately uses the correct −α D + 2κ0 D^2 form to obtain the same thresholds and asymptotics, so the final claims agree with the paper’s results and standard LT literature. See Theorem 3.3 and the inequalities (3.1), (3.5)–(3.7) in the paper for the explicit statements used here .