We establish finite-step probabilistic upper bounds for the contraction ratios represented by $\rho _{k} = \Delta _{k+1}/\Delta _{k}$ arising in iterated Pearson row–row correlation dynamics. Let $(P_{k})_{k\ge 0}$ denote the sequence generated by the Pearson correlation update, with increments $\Delta _{k} := \|P_{k+1}-P_{k}\|_{F}$, ratios $\rho _{k} := \Delta _{k+1}/\Delta _{k}$ $(\Delta _{k}>0)$, and normalized step size $\delta _{k} := \Delta _{k}/n$. Although $\Delta _{k}\to 0$ along convergent trajectories, finite-step ratios may exceed unity, a phenomenon not captured by local linearization analyses.
For fixed matrix dimension n and under the probability measure $\mathbb{P}$ induced by random initialization of $P_{0}$ with independent and identically distributed uniform $[-1,1]$ entries, we construct explicit state-dependent bounds $B_{p}:\mathbb{R}_{+}\to \mathbb{R}_{+}$ in the post-transient regime $k\ge 2$. These bounds are piecewise-constant functions $B^{\mathrm{q}}_{p}(\delta )$ obtained as empirical conditional p-quantiles of $\log \rho _{k}$ given $\delta _{k}$ under logarithmic binning. Deterministic enlargements are introduced via uniform multiplicative adjustments, yielding pointwise larger families while preserving the learned δ-dependence.
Independent validation confirms that the constructed bounds satisfy $\mathbb{P}(\rho \le B_{p}(\delta )) \ge p$ with empirical coverage matching nominal levels across $n\in [3,2000]$. Analysis of the baseline empirical 0.95-quantile bound shows $\mathbb{P}(\rho \le 1 \mid \delta \le 0.03) \ge 0.95$ for all tested dimensions, and $\mathbb{P}(\rho \le 1.7) \ge 0.95$ for 21 of 22 dimensions. These results provide the first finite-step probabilistic control for this nonlinear normalization map.
