Fixed point of a 32-step trace-normalized non-linear matrix flow

Question

Does the following 32-step non-linear iterative loop on an 8 × 8 matrix \(S\) correspond to a known invariant or classification limit in non-linear dynamical systems?

\(S_{n+1}=\frac{S_{n}S_{n}^{T}}{\text{Tr}(S_{n}S_{n}^{T})\cdot K}\)

When the baseline matrix configuration is scaled by a factor related to \(\gamma \approx 0.577215\), this discrete flow completely eliminates initial fluctuations and freezes precisely into the following numerical trace invariant:

\(\rho \equiv 0.210108\)

Are there any known algebraic theorems or matrix model literatures regarding the exact convergence boundaries of this specific discrete matrix evolution?