Stochastic Histories at Finite Temporal Resolution: Homological Observability and Information Phase Diagrams in Markov Jump Processes

Abstract

Even when all transition rates are known, the same sequence of noiseless state snapshots can arise from paths with different net windings around network cycles. Finite temporal resolution thus leaves residual randomness in the history conditioned on the record. As the observation time grows, how rapidly must the sampling interval shrink to recover the accumulated winding? For stationary Markov jump processes on finite, connected, simple networks with positive bidirectional rates, we construct an integer hierarchy of unresolved cycle differences from short-time path probabilities and integer relations. A competition of exposure order m accumulates on the scale λm = Tδm−1, where T and δ are the dimensionless observation time and regular sampling interval. When hidden cycles are present, we prove that all accumulated windings are asymptotically exactly recoverable in the joint long-time, high-frequency limit if and only if the earliest hidden layer’s exposure vanishes, requiring a superlinear snapshot budget. Under power-law sampling, the leading logarithmic coefficient of the residual conditional entropy is piecewise linear in the sampling exponent; its slope changes reveal the number of new independent cycle directions at each level. These transitions persist under detailed balance at zero steady-state entropy production rate, showing that the homological observability boundary is jointly determined by recording resolution and the growth of the history itself. Read and discuss: alphaXiv Companion code: GitHub repository · Archived software v1.1.0