Cluster Representation of Renormalization Group Transformations and a Rigorous Proof for Convergence of the RG-Flow of the Ising Model to Trivial Fixed Points away from Criticality

What the math gives to ML

The paper gives a constructive class of symmetry-preserving, probabilistically valid, monotone block transformations for Ising spins, together with an RG interpretation in which repeated coarse-graining follows a low-dimensional dynamical flow toward trivial fixed points away from criticality. The transferable asset is not the lattice-specific theorem itself, but the combination of local stochastic pooling, global Z2 symmetry, and order preservation, which prevents arbitrary learned downsampling from destroying phase-like or sign-like structure. A practical neural analogue is a stochastic RG pooling layer whose probability table is constrained by these axioms and whose single free parameter can be learned or scheduled. This is especially appropriate for CNNs, graph neural networks, and multiscale world models where representations should retain stable coarse variables while reducing spatial resolution.