Information-Efficiency of Local Observers for Distinguishing 2D Ising and Potts Configurations

We compare two observer-count scales for distinguishing sampled configurations from local observations: an adaptive-greedy threshold and an information-theoretic birthday-bound threshold. 

Defining the efficiency ratio \( \eta = K^*_{\rm bday} / K^*_{\rm greedy} \), we observe a three-regime pattern across 2D Ising and Potts models: redundant observations (\( \eta < 1 \)) at low temperature, near-efficient criticality (\( \eta \approx 1 \)), and complementary observations (\( \eta > 1 \)) at high temperature. 

A conditional conjecture under polynomial alpha-mixing suggests \( \eta \to 1 \) at \( T_c \) for FK(\( q \)) with \( q \in \{2,3,4\} \).