Memory Retention and the Classification of Renormalization-Group Fixed Points in Self-Similar Dynamics

What the math gives to ML

The paper introduces a renormalization-group classification based on whether an initially relevant length scale is erased or remains explicitly encoded in the asymptotic fixed point. Its transferable mechanism is a scale transformation that preserves a latent scale variable rather than forcing all states into a universal, scale-free profile: memory-retaining fixed points take forms such as \(\eta^{\alpha}F(\xi/\eta^{\beta})\). A neural implementation should combine coarse-grained feature processing with an explicit scale state \(\eta\), and train the network to predict how representations transform when the input is rescaled. The key falsifiable test is whether a nonzero learned \(\beta\) and retained-scale model produce lower cross-scale prediction error than a conventional scale-invariant or scale-blind architecture.