Compressing the Division Table into a Single Dynamical Column

Abstract. The odd division table rewritten as one time-ordered series, with nothing lost. Each odd prime is a clock starting at p^2 and stepping by 2p; each integer becomes a vector of prime-power layers whose inner product is log gcd; time averages give a Dirichlet convolution of the von Mangoldt function, and the Dirichlet transform gives the prime side of -zeta'/zeta. Three spectral objects built from it are tested and all three close for structural rather than numerical reasons.