Positional Bit Bias in Double SHA-256 and Implications for Proof of Work Systems

The SHA-256 cryptographic hash function is widely assumed to produce uniformly random outputs, satisfying the Strict Avalanche Criterion (SAC) where each output bit changes with 50% probability when any input bit is flipped. This paper presents empirical evidence challenging this assumption. Through a novel "smear sampling" methodology testing over 1.1 million hash operations, we identify persistent positional bit biases in both single and double SHA-256 implementations. Specific bit positions (notably 108, 95, 252, 110, 68, and 7) consistently appear among the most statistically extreme positions across multiple sample sizes. These findings have implications for proof-of-work systems like Bitcoin, where selection effects may amplify subtle biases in filtered hash outputs.