Regulated Accumulation in Dynamical Systems

Dynamical systems exhibit divergence through long-time iteration, spectral proliferation, and uncontrolled accumulation of orbits, measures, and modes. ^[7][10] Classical responses; symbolic truncation, coarse graining, or probabilistic renormalization; address outcomes rather than structure. ^[7][8][9] This paper applies the Additive–Multiplicative (AM) Regulator as a fixed mathematical framework to dynamical systems, constraining accumulation itself without altering the dynamics. Regulated accumulation induces spectral saturation, compactness at fixed constraint, and controlled regime change. ^[2][3] Classical chaotic and ergodic behavior is recovered exactly as a degeneracy limit. ^[7][10] The conclusion is direct: divergence in dynamical systems is not intrinsic to dynamics, but a consequence of unconstrained accumulation.