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 \( η = K^*bday / K^*greedy \), we observe a three-regime pattern across 2D Ising and Potts models: redundant observations (\( η < 1 \)) at low temperature, near-efficient criticality (\( η ≈ 1 \)), and complementary observations (\( η > 1 \)) at high temperature. A conditional conjecture under polynomial alpha-mixing suggests \( η → 1 \) at \( T_c \) for FK(\( q \)) with \( q ∈ \{2,3,4\} \).