We present the discovery of a fundamental composition law governing conjugate observables in the Random Permutation Sorting System (RPSS). The law links the discrete permutation count Np and the continuous elapsed time T through a functional relation connecting the characteristic function of timing distributions to the probability generating function of permutation counts. This framework enables entropy purification, transforming microarchitectural timing fluctuations into uniform randomness via geometric convergence. We establish convergence theorems with explicit bounds and validate the results experimentally, achieving Shannon entropy above 7.9998 bits per byte and chi-square uniformity across diverse platforms. The composition law provides a universal foundation for generating provably uniform randomness from general-purpose computation, securing cryptographic purity from emergent computational dynamics.

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