Chronological stability for nonstationary matrix refinement

A matrix cascade is an iterative refinement process in which vector-valued data are repeatedly filtered by matrix-valued masks across successively finer scales. Such cascades arise in multichannel subdivision, Hermite refinement, multiresolution synthesis, and exponential-spline constructions, and in nonstationary schemes the masks may vary with scale. We develop a chronological stability framework for this setting, allowing both the transition operators and the associated matching spaces to change from level to level. If the chronological radius satisfies ${\rho }_{\mathrm{c}\mathrm{h}}<1$ and the matching defects are summable, then the cascade converges uniformly and, for every $q\in ({\rho }_{\mathrm{c}\mathrm{h}},1)$, $$\mathrm{\parallel }{F}_n-\mathrm{\Phi }{\mathrm{\parallel }}_{\mathrm{\infty }}\lesssim q^n+\sum _{j=1}^n{\delta }_j{q}^{n-j}+\sum _{j>n}{\delta }_j\mathrm{.}$$ The time-weighted defect term records the level at which each perturbation is inserted, and the matching spaces need not converge; they may even alternate indefinitely. For compact piecewise-linear seeds, every finite ordered matrix cascade admits an exact fixed-width ReLU realization with $O(n)$ depth and parameters, and this linear depth order is worst-case optimal at fixed width. Consequently, exponentially decaying defects yield certified approximants of size $O(\log {\epsilon }^{-1})$. We further construct stable two-channel classes in which the chronological radius and the leading metric-entropy coefficient are independent invariants, showing that stability does not determine finite-accuracy description complexity. The framework therefore provides tolerance-controlled finite representations for nonstationary multiresolution and exponential-spline refinement.