Reference-Weighted Deterministic Quantum Flows with Singular Interactions and Retained Sources

Under what conditions does a quantum probability current determine complete, deterministic configuration histories, including nodes, singular interactions, and dynamical source coordinates? This paper establishes a theorem framework connecting quantum continuity equations to deterministic flows under explicit regularity, integrability, and boundary assumptions. The main result proves avoidance of wave nodes throughout a finite time interval and deterministic selection of reference-compatible paths. Uniqueness is established within a specified class of path laws whose time marginals are bounded by a finite multiple of the reference density. Any original configuration law absolutely continuous with respect to the initial wave density is then transported by reweighting those same paths, without introducing a new statistical population during evolution.The paper also proves approximation and stability results for entire histories. These require control of both the complete density and current, and explicitly account for trajectories leaving the region where the comparison estimates apply. Applications include discontinuous radial interactions with finite spin and quantum sources, bare Coulomb interactions, and a Coulomb electron coupled to a retained quantum oscillator. A separate theorem bounds endpoint and crossing errors for suitably regular comparison currents with a signed continuity defect. Counterexamples identify failures caused by omitted source currents, singular initial laws, and insufficient control near nodes. These results provide mathematical foundations for deterministic quantum transport and record analysis. They do not select an initial Born distribution or independently establish a complete measurement apparatus.