Sharp estimates for Toda-type fully bubbling sequences of the $ \mathrm{SU}(3) $ Chern–Simons system on a flat torus

We derive sharp asymptotic estimates for Toda-type fully bubbling sequences of the $ \mathrm{SU}(3) $ Chern–Simons system on a flat torus. We assume that the blow-up set consists of finitely many regular points and impose a critical relation between the local amplitudes and the bubbling scales. At each blow-up point, we select an entire Toda solution from the full eight-parameter family. The proof uses a weighted estimate for the linearized Toda operator on expanding disks modulo its eight-dimensional Jacobi kernel. Projection onto the eight Jacobi fields yields a second-order local mass expansion, location identities for both components, and an identity coupling the trace-free Hessian of the difference of the limiting effective backgrounds to an explicit Chern–Simons coefficient of the limiting Toda profile. For each fixed cutoff, we identify the subsequential limits of the normalized local mass defect and determine their dependence on the cutoff radius. The Chern–Simons coefficient maps arising in these projections are real-analytic and not identically zero.