We exploit the properties of complex time to obtain an analytical relationship based on considerations of causality between the two conserved quantities of a system’s Hamiltonian and its Entropy Production. In natural units, when complexified, the one is simply the Wick-rotated complex-conjugate of the other. A Hilbert transform relation is constructed in the formalism of Quantitative Geometrical Thermodynamics which enables system irreversibility to be handled analytically within a framework that unifies both the microscopic and macroscopic scales, and which also unifies the treatment of both reversibility and irreversibility as complementary parts of a single physical description. In particular the thermodynamics of two unitary entities are considered: the alpha particle which is absolutely stable (that is, trivially reversible with zero entropy production), and a black hole whose absolute disequilibrium (unconditional irreversibility) is characterized by a non-zero entropy production for which we show an alternate derivation confirming our previous one (Universe 7, 2021, 325). The thermodynamics of a canonical decaying harmonic oscillator are also considered. In this treatment the complexification of time also enables a meaningful physical interpretation of both “imaginary time” and “imaginary energy”.