Constrained Walks on de Bruijn Graphs and the Dynamics of Exhaustion Systems

We introduce the model of Deterministic Games with Irreversible Global Memory (DGIGM), a class of constrained systems in which every action irreversibly consumes a portion of the future action space. We prove that the constraint structure of a DGIGM with memory depth K is exactly isomorphic to a non-repeating edge walk on the de Bruijn graph B(n,K), where n is the number of valid actions. The simulator Ω-TRACE implements a DGIGM coupled to a two-dimensional geometric space, creating a dual system in which an abstract combinatorial constraint coexists with concrete physical constraints. We analyze the system’s properties: the phase transition in maneuverability, the Efficiency Paradox (where local optimization accelerates global collapse), and the emergence of complex morphological structures documented by a catalog of 425 unique forms extracted from477 game sessions. We propose the DGIGM as a benchmark for evaluating the ability of artificial agents to operate under non-renewable resource regimes, and formulate a testable conjecture on the superiority of distributed strategies over optimizing ones.