Frozen Calogero–Moser–Sutherland dynamics are transformed, in the asymptotic regime of high inverse temperature, into deterministic particle dynamics of reciprocal, boundary, interval, and cotangent type. Near-collision initial conditions make the explicit integration difficult since the particle variables keep their denominators, namely, $(x_i-x_j)^{-1}$ and $(x_i+x_j)^{-1}$, throughout the initial layer of separation. Residue-cancellation-based polynomial integration (\RCPI) computes the coefficients of the monic root polynomial instead of computing the singular drift explicitly at each step of integration. The computation is done for the Hermite, Laguerre, compact Jacobi, non-compact Jacobi, and torus cases, employing four six-particle starts at $T=0.35$ and initial separations down to $4\times10^{-4}$. The finite-dimensional system of coefficients arises from the inverse heat, inverse Bessel-heat, inverse Jacobi-type, and torus operators. The roots are validated by maximum root discrepancy, closure residual, radial growth, interval confinement, and circular spacing. In the case of the Hermite, Laguerre, compact Jacobi, and torus computations, the discrepancy of the final roots from the direct singular ODE integration falls between $5.11\times10^{-15}$ and $3.06\times10^{-14}$ while the evaluations of the right-hand side have been saved by a factor ranging from $3.02$ to $23.80$. The values demonstrate that the potential near-collision singularity is stripped away from the time integration process once the residue cancellation is stated in terms of the polynomial coefficients.

Calogero--Moser--Sutherland systems,root polynomial dynamics,freezing limits,inverse heat operators,coefficient propagation,near-collision