Mathematical modelling of a multicluster open system: transfer-operator recovery from aggregate data under a least-action prior

Many open systems reveal themselves only through aggregate snapshots of their states, while the rule that moves objects between states stays hidden. This paper studies the inverse problem of recovering the transfer operator of a multicluster open system from two adjacent snapshots. The problem is severely under-determined. The operators consistent with the data fill an affine manifold, so the data fix the aggregate flows and leave the internal routing free. Every recovery method commits to a prior that selects one operator from this manifold. Tikhonov regularisation with a zero reference selects the minimum-norm operator, which spreads outflow as evenly as the constraints allow and underestimates how much of each cluster stays in place. For an inertial system this default is wrong. We read the choice of reference as a substantive a priori condition and propose least action on the operator trajectory, under which the rule of evolution changes as little as the data permit (informally, “behavioural laziness”). We separate the inertia of the state, which equals the naive forecast, from the inertia of the rule, which carries the signal the naive forecast misses. A causal construction keeps the observed balance exact by moving the operator only inside the null space of the constraints. We make precise the links to optimal transport and to the Fokker-Planck / Perron-Frobenius coarse-graining, and relate the construction to the Schrödinger bridge. Two instantiations of one model carry the method: the Russian labour market and a radiation-damped relativistic electron ensemble. The null-space construction holds the balance to machine precision where temporal smoothing corrupts it by about twelve percent on the labour series and about sixty percent in the physical one. In the physical instantiation the spectral gap is non-identifiable from aggregate snapshots and available from micro-trajectories, where Ulam’s method reproduces the Landau-Lifshitz relaxation rate to within five percent on trajectories from the known model. On the short labour series no method beats the naive forecast in point accuracy at any conventional significance level. The value of the approach is structural. It recovers an interpretable operator while preserving the observed balance exactly, and it marks the information limit of aggregate observation.