Renormalization for Robust Memory without Synchronization

Local error correction can preserve classical information against noise, but protection under synchronized updates need not survive when sites evolve at independent random times. Establishing stability is particularly challenging when protection relies on the collective action of competing local rules. Here we develop a spacetime renormalization framework that reveals how independently acting local updates can collectively stabilize memory. We identify sufficient conditions under which the geometry of stable interfaces and suppression of defect propagation produce reliable erosion at large scales. Applying this framework, we prove stability of the asynchronous \textsf{R} squeezing code and introduce robust processes in two and three dimensions. In these systems, both encoded phases persist under sufficiently weak noise, even when the noise favors one phase. Our results provide a criterion for designing autonomous memories sustained by competing local stochastic dynamics.