A Hard-Core Subshift Whose Sofic Mean Dimension Depends on the Sofic Approximation

What the math gives to ML

The paper gives a constructive example showing that the effective continuous dimension of a locally constrained system can depend strongly on the graph approximation used to represent the same nonamenable dynamics. Its hard-core constraint forbids simultaneous nonzero values on adjacent sites, so achievable dimension is controlled by the independent-set structure of the action graph: bipartite approximations attain dimension 1/2, while random-permutation approximations attain a value between 1/5 and 9/20. A transferable neural mechanism is graph-approximation-aware sparse routing, where local incompatibility constraints restrict simultaneously active experts, neurons, or token groups. The key experiment is to hold the local rule fixed while changing only the interaction graph and test whether active capacity and optimization behavior exhibit the predicted separation.