Distribution Analysis of Discrete States: reconstructing transition processes from time-resolved state distributions

<title>Abstract</title> <p>Many scientific and applied domains repeatedly observe populations or systems and record how objects are distributed across discrete categories. This paper presents Distribution Analysis of Discrete States (DADS), a general framework that treats any observation matrix N(state, time) as a flow network and reconstructs the underlying transition processes from observed state distributions. The framework rests on three foundations: a Level 0 network topology decision, made before any other modelling choice, that fixes the dimensionality of the null space separating observed from true transition rates; an extended δ-typology distinguishing four sources of non-identifiability (physical, semantic, epistemic and structural); and a formal treatment of observational completeness that separates census data from sample data and their different implications for the estimator family (V0.1–V0.5). A three-term null-space decomposition (state-circulation, bipartite-marginal and external-anchor components) is derived and cross-checked against two established frameworks. Seven case studies spanning sociology, electoral politics, materials science, business demography and fisheries apply the framework and its diagnostics (state and transition entropy, a coupling index, the Jensen-Shannon divergence, and the Earth Mover’s Distance for ordered states). A block-bootstrap null test applied post-hoc to one diagnostic illustrates the framework’s self-correcting character: an initially reported cross-domain consistency result proves, on closer testing, indistinguishable from sampling noise and is withdrawn.</p>