Coherence, Circulation and the Quantum Regime

The completed Chronoflux Atlas recovers the quantum regime as the coherent phase branch of conserved temporal transport. Continuity supplies the current, temporal order and positive kinetic response recovered in DPL-00 through DPL-04. Coherence adds a non-vanishing complex state with a physical phase and action scale. Once that descendant is admitted, variation returnsphase transport, topology quantises circulation, the Fisher-gradient term returns the Bohm quantum potential, and the density-phase equations recombine exactly into the Schrödinger equation. Quantum behaviour is therefore not attached to continuity by analogy. It is recovered as the lawful dynamics of a coherent continuity branch. The coherent action gives a conserved density current and a quantum Hamilton-Jacobi equation. On the domain where the amplitude does not vanish, the phase one-form is locally exact. Its global periods are integer winding numbers, giving circulation in units of ℎ/𝑚. Those integers persist under every continuous admissible evolution and can change only when a defect crosses the return path or the coherent amplitude vanishes. The apparent discontinuity of a quantum phase slip is thus the topological consequence of continuous evolution through a configuration where the phase ceases to be defined. The same structure returns the exact Madelung representation of a selected Schrödinger branch. On configuration space it yields the Born current, current velocity and quantum potential. Adding an actual configuration following that velocity is the Bohmian ontological completion, not an unstated consequence of continuity. Equivariance then proves persistence of quantum equilibrium. At field level, a shift-symmetric action closes a classical current, its quadratic perturbations recover a characteristic metric, and stationary canonical data produce a recoverable bosonic quantum state. Gauge selection, spin, fermionic statistics, outcome selection and Standard Model promotion remain separately indexed descendants. The paper also closes the non-ideal boundary. Multiplying a density by an arbitrary positive weight generates an exact continuity source and an exact transformed quantum potential; it cannot be presented as a free admissibility correction. Any departure from the closed branch requires a declared self-adjoint operator or trace-preserving open-system law. Admissibility, Recoverability and Persistence then distinguish a genuine quantum branch from incomplete phase data, singular nodal flow or an unreturned measurement model. The result closes coherence, quantised circulation and quantum response within the recovered tree of nature as such.