Sinkhorn-Knopp doubly-stochastic normalization at a fixed iteration count wastes compute on dense matrices and under-serves sparse ones, measured directly.
Sinkhorn-Knopp (SK) is the standard way to push a matrix of positive numbers toward doubly stochastic: alternately normalize every row, then every column, and repeat. It shows up across manifold-constrained hyper-connections (mHC, arXiv:2512.24880) and other neural-network mixing schemes as a fixed 20-iteration budget. This gist measures what that fixed budget actually buys, at 5/10/20/50 iterations, on two synthetic matrix regimes chosen to sit on either side of the density phase transition He (arXiv:2507.09711) identifies for Sinkhorn-Knopp convergence. In CCF, SK-style normalization is used only as a gauge/presentation step, never as the causal trust-transfer update itself, which is a separate quotient-affine contraction (QAC) step. This