<div> We develop a structural theory of monetary circulation under partial observability. Money is modeled as a path-dependent semi-Markov process moving across an institutional network, where transfers are observed only at aggregated endpoints while part of the routing occurs through hidden or unrecorded channels. Rather than at- tempting to reconstruct unobserved flows, we characterize their observable imprint on the visible economy. We show that hidden circulation induces three robust ef- fects: (i) distortions of measured turnover and residence times, (ii) endogenous trapping and congestion not ex- plained by network topology alone, and (iii) systematic irreversibility that may remain invisible in observed flows. The key object is a phase field defined on visible links, inter- pretable as a directional shadow price of hidden circulation, orthogonal to incentive-based (potential) explanations. Irreversibility is formulated at the level of monetary paths via path-space relative entropy. We prove that entropy pro- duction in the observed economy is always a lower bound on true entropy production, with a nonnegative gap captur- ing hidden irreversibility. When the hidden phase contains a genuine cycle component and the economy uses that cy- cle, hidden entropy production is strictly positive even if observed circulation appears nearly reversible. We connect these structural results to macroeconomic out- comes by defining a price-free structural output proxy based on the interaction of monetary activation and directional circulation. Under stable institutional regimes, this proxy is shown to be affine-comparable to measured GDP growth. The framework provides a unified account of dark circulation, irreversibility, and macroeconomic inefficiency, without invoking equilibrium prices, representative agents, or welfare optimization. This paper completes a three-paper sequence by extending the framework to hid- den and partially observed circulation. </div>