The Incompressibility Limit of the Prime Sequence

We propose a new theory of structural complexity for the prime number sequence, based on a residual compressibility metric λk. By comparing compression ratios of prime sequences and their modeling residuals under various structure-based approxi- mations (e.g., log x, Li(x), Riemann R(x)), we observe that the prime sequence exhibits a compressibility pattern that approaches but never reaches that of true random se- quences. This suggests that primes represent a unique class of boundary structures — neither fully random nor fully ordered — which we formally describe as the Prime Residual Compressibility Limit (PRCL).