the proof

the mathematics that the protocol is computing.

in 1965, Andrey Kolmogorov defined the complexity of a finite binary string as the length of the shortest computer program that produces it on a universal turing machine. the definition is independent of the specific machine used, up to a constant. a string that has a short program describing it is simple. a string with no short program is random.

the complexity of a string of length n is at most n plus a small constant, because any string can be described by a program that just prints the string. the interesting case is when the complexity is much smaller than n. that is when structure is present.

the inequality

for almost all strings of length n, the kolmogorov complexity is close to n. this is a counting argument. there are two to the n strings of length n, but only two to the n minus c short programs of length n minus c, so most strings have no short description.

the consequence for markets is direct. if the trade stream of an unmanipulated token is approximately random, its kolmogorov complexity should be close to the length of the stream itself. anything significantly shorter is evidence of hidden structure.

the approximation

kolmogorov complexity is uncomputable in the strict sense, because finding the shortest program for a given string is equivalent to solving the halting problem. but the complexity can be approximated from above by any specific compression algorithm. the size of a string after being compressed is an upper bound on its true kolmogorov complexity. and a tight upper bound is enough to detect when a string has become compressible.

the protocol uses lempel-ziv 77 as its compression algorithm, modified for streaming evaluation inside solana's compute budget. the algorithm has the property that for any string of length n produced by an ergodic source with entropy h, the lz77 encoding length divided by n approaches h as n grows. this is the connection to information theory.

the natural reference

the protocol's reference value of 0.853 is the empirical kolmogorov complexity, measured by the lz77 approximation, of a trade stream generated by a simulated population of independent traders acting on private signals. the value was computed by running the simulation for one million synthetic transfers and measuring the average normalized compression ratio. it is hardcoded into the binary. it cannot be changed by anyone.