A question common to quantum information, complexity theory, mathematical physics, and random matrix theory is how locally random quantum processes converge toward globally random dynamics. Quantum pseudorandomness and k-designs quantify how ensembles of local random unitaries resemble global random unitaries. A key consequence of convergence to k-designs is the ‘incompressibility’ of random circuits [Chen, Haah, Haferkamp, Liu, Metger & Tan 2024]: the minimum size of any approximating circuit typically grows linearly in the original circuit’s depth until exponentially large in the qubit number, albeit with conjectured polynomial inefficiencies in qubit number. Works of Araiza, Chen, Ding, Junge, Schleich, and Wu [2024 & 2025] connect decay rates under quantum Markov semigroups to their simulation complexity. We bridge these approaches to show that entropy produced under k-fold collective mixed unitary channels, which apply the same randomly drawn unitary to k systems, lower bounds the typical circuit complexity of constituent unitaries. We further derive a relative entropy decay recombination theorem relating decay rates of differently structured processes. Combined with recent works [Schuster, Hakerkamp & Huang 2024. L & Leditzky 2024] using random matrix theory and von Neumann subalgebras to construct efficient designs, this theorem tightens random circuit incompressibility to within polylogarithmic factors of the best possible.