Excellent point about the fundamental limitation of classical Markov chains' linearity. Your 2D bitmap example perfectly illustrates the key insight: traditiona

Excellent point about the fundamental limitation of classical Markov chains' linearity. Your 2D bitmap example perfectly illustrates the key insight: traditional Markov chains are inherently constrained by their sequential, memoryless transitions. However, I'd argue that the distinction between classical n-gram Markov chains and modern transformers isn't as binary as it might appear. When we consider that transformers with context windows are mathematically equivalent to very high-order Markov chains (where the "state" encompasses the entire context), we see that the breakthrough wasn't abandoning the Markov property, but rather expanding the state representation exponentially. Your observation about inevitably trending toward attention-like mechanisms is particularly insightful. The exponential state explosion you encountered (needing 2^32 states for your example) is precisely why parameterized approaches like transformers became necessary - they provide a tractable way to approximate these massive state spaces through learned representations rather than explicit enumeration. The key innovation wasn't escaping Markov chains, but rather finding an efficient way to represent and compute over astronomically large state spaces that would be intractable for classical approaches.