History Cohomology and the Memory of Assembly: Path Observables, Krieger Types, and the Dressing of Time

How much information survives about how something was built, even after you know exactly what it became? This paper develops a mathematical framework for that question. Rather than treating history as a sequence of events, it treats histories themselves as geometric objects whose topology can be analyzed. The resulting theory, history cohomology, distinguishes information that can be recovered from a final state from information that is fundamentally encoded in the path taken to reach it. In this framework, "assembly memory" becomes a precise mathematical object rather than a philosophical metaphor. The paper proves several structural results, including a rigidity theorem showing that genuine assembly memory cannot hide beyond arbitrarily deep finite approximations under broad conditions. It then applies the framework to graph reconstruction, identifying concrete observables that either forget or permanently remember how a graph was assembled. These ideas are extended into probability, ergodic theory, and operator algebras, where the same mathematical distinction predicts when growth processes erase their construction history and when that history survives indefinitely. The work also introduces a categorical "dressing" principle showing that assembly memory behaves like a conserved quantity: it cannot simply disappear, only be relocated into additional boundary information. This provides a common mathematical language connecting graph growth, categorical topology, measurable dynamics, modular operator theory, and notions of information preserved across scales. The broader implication is a shift in perspective. Instead of asking only whether two final states are equivalent, this framework asks whether they are equivalent as products of history. That distinction has potential consequences for graph theory, stochastic growth processes, reconstruction problems, information theory, distributed systems, and any domain where the route to a final configuration may itself carry irreducible information. Whether these ideas ultimately find applications in mathematics, physics, or computation, the central claim is that assembly history admits a natural topological theory—and that this theory reveals phenomena invisible to endpoint-based descriptions alone.