Conditions for Unguided Reachability of Guidance-Like Computation

Key Findings

• Exponential Decay Conjecture: The fitted model log P(hit) = -0.0108 * I_eff - 2.306 shows a general downward trend (R-squared = 0.287), but the moderate fit quality suggests the decay is not purely exponential. Hit rates plateau around 0.04--0.05 for guidance lengths L >= 6.
• Temperature Controls Exploration: The spectral gap increases monotonically from 0.00024 (tau = 0.3) to 0.3045 (tau = 10.0), spanning over three orders of magnitude. Higher temperature improves mixing but does not change the stationary target mass, which remains constant at approximately 0.259.
• Curriculum Viability: The curriculum approach maintains all stage hit rates above 0.559 (minimum at stage k=7). Per-step information alternates between low-information steps (1.538 nats) and high-information steps (4.038 nats), with a mean of 2.788 nats.
• Phase Transition in State-Space Size: The parameter sweep reveals that reachability collapses for larger state spaces. At (D=5, B=4) with 1024 states, the hit rate drops to 0.020 at tau=0.5. Configurations with I_eff above approximately 40 nats tend to have hit rates below 0.2.
• Practical Diagnostic: If the estimated effective information of a guidance prefix exceeds roughly 40 nats, unguided reachability is unlikely. The spectral gap should be above approximately 0.001 for practical exploration.