  The continuous min flow-max cut principle is used to reformulate the
'complexity=volume' conjecture using Lorentzian flows -- divergenceless
norm-bounded timelike vector fields whose minimum flux through a boundary
subregion is equal to the volume of the homologous maximal bulk Cauchy slice.
The nesting property is used to show the rate of complexity is bounded below by
"conditional complexity", describing a multi-step optimization with
intermediate and final target states. Conceptually, discretized Lorentzian
flows are interpreted in terms of threads or gatelines such that complexity is
equal to the minimum number of gatelines used to prepare a CFT state by an
optimal tensor network (TN) discretizing the state. We propose a refined
measure of complexity, capturing the role of suboptimal TNs, as an ensemble
average. The bulk symplectic potential provides a 'canonical' thread
configuration characterizing perturbations around arbitrary CFT states. Its
consistency requires the bulk to obey linearized Einstein's equations, which
are shown to be equivalent to the holographic first law of complexity, thereby
advocating a notion of 'spacetime complexity'.
