There is an invisible hand mechanism in the property system that underlies the invisible hand mechanism in the price system. In the life-cycle of property rights, initiation-transfers-termination, the invisible judge imputes the initial rights and terminal liabilities according to the public part of the life-cycle, the contractual transfers. If the legal system does not intervene, then the invisible judge laissez-faire imputes the termination of a property right to the last buyer and the initiation of a right to the first seller. When the legal system does intervene to hold a trial, it attempts to implement the principle of imputing de jure responsibility in accordance with de facto responsibility (the juridical version of the Lockean fruits of one's labor principle). Hence the natural question is: under what conditions does the invisible judge satisfy the responsibility principle? Hume emphasized two basic conditions: that all transfers in property be voluntary contracts and that all contracts be fulfilled. The fundamental theorem for the invisible hand mechanism in the property system is that if Hume's conditions are satisfied, then the invisible judge imputes in accordance with the Lockean responsibility principle. The paper mathematically formulates and proves the theorem using vector flows on graphs. The penalties used to enforce Hume's conditions have a duality theory which is outlined as a limiting case of price-theoretic duality.