A Reconstruction Protocol and a Hierarchy of What It Recovers
Many fields fit “landscapes” to metastable time series: ecology, neuroscience, cell biology, finance, collective behaviour. The fitted object is rarely defined precisely, and the question of which of its features survive the analyst’s choices is rarely asked. We define the landscape as an empirical object — the effective potential Veff = −D log Pss obtained from the stationary density — state the conditions under which it coincides with a modeled potential and why it governs transition times even when it does not, and specify a six-step reconstruction pipeline. We then validate the assembled pipeline blind on simulated trajectories, recovering barrier heights to within two percent and predicting absolute dwell times to a median error of three percent with no fitted parameters. The central organizing result is a hierarchy of recoverable quantities, established by running the same data through deliberately different normalizations: the shape of the dwell-time distribution needs only a state label; barrier heights are invariant under any fixed linear change of coordinates; absolute rates are invariant under nothing, and the sign of their error flips between whitening (+20%) and z-scoring (−26%). Geometry is a property of the system; kinetics is a property of the system and the coordinates through which it is observed. Two aggregation artefacts are quantified and corrected. Pooling dwell times across units with different barrier heights gives a heterogeneous population of perfect Poisson escapers rejected as non-exponential in 98.9% of samples; rescaling by each unit’s own mean restores the nominal rate. Shortening the analysis epoch to resolve a drifting landscape recovers exponentiality as intended, but the same operation manufactures it in a stationary control once epochs hold fewer than about ten mean dwell times. Applied to published resting-EEG measurements, the protocol yields a quantitative constraint: smooth barrier drift saturates at β≈0.74 and cannot account for the observed β≈0.5, which points to scale-free rather than bounded landscape motion.
Keywords — potential landscape, quasipotential, Boltzmann inversion, Kramers escape, metastability, dwell-time distributions, reconstruction, coordinate invariance, critical transitions