# Section 1: From a quantum current to a complete history
## 1 From a quantum current to a complete history
A quantum density and a continuity equation do not by themselves specify a deterministic history through nodes, collision sets and retained source coordinates. The question is sharper when the original configuration law need not equal the wave density: one needs a flow for that original law, without inserting a new distribution at a later preparation or observation time. The same issue appears in approximation. A small wave error does not directly control an entire configuration path, and a small contracted current can hide large opposing currents at different source configurations.
This paper develops a precise route from complete quantum currents to deterministic reference-compatible histories. A superposition theorem first supplies integral-curve path laws with the wave-density marginals. Finite action, logarithmic variation, local positive-tube Sobolev regularity and separate boundary control then make the conditional path law at almost every entrance a point mass. The uniqueness class consists of integral-curve path laws dominated at every time by one finite multiple of the reference density. It is not a claim about every ordinary differential equation solution from every exceptional point. Every original law absolutely continuous with respect to the entrance density is transported by reweighting the same reference paths once; a finite density cap is required only for the stated quantitative transfers.
We next prove two complementary approximation results. Strong convergence of both density and complete current, together with uniform action and a fixed entrance, identifies weak limits of entire path laws without asserting convergence of a velocity quotient near nodes. A logarithmic comparison on a common positive tube then bounds the probability that two genuine flows separate anywhere during the full interval. The price includes the probability of leaving that tube. Discarding the bad-tube term, or replacing the complete current by a positional marginal, changes the theorem.
The applications retain features that simple smooth-potential presentations often remove: discontinuous radial interactions with finite internal spin, a bare Coulomb collision with an uncut entrance, and a Coulomb electron coupled to a dynamical oscillator source. The regularity arguments distinguish common first domains, tangent commutators and product radial–tangent norms from unavailable high powers of the full Hamiltonian. A separate compact-entrance construction supplies classical flow charts where a fixed-surface crossing argument requires them. Finally, Appendix [A](/quantum-measurement/research/reference-weighted-flows/appendix-a-approximate-continuity-and-a-whole-path-source-allowance#p3r:signed-source) gives a signed-source comparison theorem for a nonconservative reference, with explicit endpoint and whole-path allowances.
### 1.1 Established theory and the contribution here
Global existence for Bohmian trajectories, including control of nodes and singular boundaries, has an established literature. Berndl and collaborators [[1](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-BerndlEtAl1995)] and Teufel–Tumulka [[2](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-TTflow)] are direct predecessors. In particular, finite trajectory length and logarithmic-density variation are established mechanisms, and Teufel–Tumulka already treat general current conditions, spin, magnetic fields and Dirac examples. We use those mechanisms with the explicitly declared complete current; they are not claimed as new principles.
The distinction between a weak continuity equation and a well-defined flow is central to DiPerna–Lions [[3](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-DiPernaLions1989)] and Ambrosio [[4](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-Ambrosio2004)]. The logarithmic estimate below uses the Sobolev maximal-function inequality of Crippa–De Lellis [[5](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-CDLflow)]. Our superposition step imports the finite-dimensional case of Stepanov–Trevisan [[6](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-STsuperposition)] with its hypotheses checked explicitly. Ambrosio–Colombo–Figalli [[7](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-ACFmaximal)] provide related local maximal-flow theory; its additional hypotheses are not silently substituted for the reference-density assumptions used here. Common-domain nonautonomous evolution likewise has established antecedents, including the account of Schmid–Griesemer [[8](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-SchmidGriesemer2014)]. Yajima [[9](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-Yajima2016)] treats related multiparticle propagator regularity. The common-form and tangent-propagation argument in Section [6](/quantum-measurement/research/reference-weighted-flows/a-first-domain-route-to-full-wave-continuity#p3d:sec:graph) is proved here; these evolution references provide context rather than replacing its domain checks.
The specific assembly proved here combines reference-weighted deterministic selection, all-times node exclusion, whole-path approximation and stability, and the stated singular/source applications. Restricting a coupled path law to an event concerning both *whole* paths preserves marginal domination; using already-stopped paths as though they had the same compression would instead introduce boundary atoms. This distinction is explicit in the proof. The canonical first-domain energy calculations supply finite path and logarithmic costs, while the applications separately address continuity, local Sobolev regularity, form approximation and physical boundaries.
The author's earlier pilot-path manuscript [[10](/quantum-measurement/research/reference-weighted-flows/bibliography#bib-RodgersPilot2026)] concerns a finite-graph hybrid Bell-path setting and conditional continuum limiting interfaces. That inherited work is not a prior proof of the continuous Sobolev-current theorem presented here. Other earlier statistical and record arguments require admitted flows but address different questions. The present results do not choose a quantum equilibrium ensemble, and their substantive mathematical value does not depend on completing a physical measurement apparatus. The contribution is the stated combination of sufficient hypotheses, proofs and applications; no claim of exhaustive priority is made.