From a complete current to a history
Reference-Weighted Deterministic Quantum Flows with Singular Interactions and Retained Sources
Complete currents determine reference-compatible histories under explicit analytic hypotheses, including applications with singular interactions and retained sources.
Jeremy Rodgers · Independent Researcher · 9 October 2026
9 October 2026 manuscript
The argument
From a complete quantum current to a complete history
The continuity equation describes how density moves. A deterministic account also needs paths that remain defined through the entire interval, with nodes, singularities, and source coordinates treated explicitly. This paper establishes that passage for a declared complete quantum current.
Finite action and logarithmic variation, local Sobolev control where the density is positive, and physical-boundary estimates select deterministic reference-compatible histories. Uniqueness holds in the stated class of path laws with a common finite marginal-domination bound. Any original entrance law absolutely continuous with respect to the wave density then travels along the same paths by one initial reweighting.
The framework also controls approximation and stability of entire histories. Its applications retain radial interaction jumps, bare Coulomb collisions, and a dynamical quantum oscillator source. A separate signed-source theorem handles sufficiently regular comparison currents with a continuity defect. The proofs make the required domains and the cost of exceptional histories explicit.
See how deterministic paths are selected
The logarithmic estimate prevents almost every admitted path from reaching a node at any time. Local Sobolev control then makes the conditional path law deterministic.
Read the argument →Transport the original law through a limit
Strong density and complete-current convergence, together with uniform action and a fixed entrance, identify the limit of entire path laws and their original-law reweightings.
Read the argument →Compare whole histories
The comparison controls the largest path separation during the interval. Its budget includes the probability that either path leaves the common comparison region.
Read the argument →Keep a quantum source in the flow
The electron and oscillator evolve as one parent, with both physical currents retained. The domain argument covers the Coulomb cusp and the source’s entrance node.
Read the argument →Handle a continuity defect
For the stated classical regularity, the signed-source allowance controls endpoint discrepancy, terminating paths, and transverse-crossing probabilities.
Read the argument →P3 supplies flow and history theorems that an application can use after matching its full parent, current, entrance law, and regularity. Its retained electron–oscillator example connects directly to P4’s compact-profile model when the profiles and units agree. The explicit preparation and receiver constructions in P1 and P2 retain their own hypotheses and quantitative budgets.
The complete paper
Follow the full argument.
Every section, proof and appendix, with linked equations and the complete bibliography.
- OpeningAbstract and publication identity
- Section 1From a quantum current to a complete history
Established theory and the contribution here
- Section 2Complete currents and reference-compatible path laws
Smooth tests, boundaries and infinity
- Section 3The logarithmic chain rule and deterministic selection
Checking the wave-to-velocity hypotheses
- Section 4Conforming approximation and the original history law
- Section 5Whole-history stability in a positive-density tube
The velocity defect uses both density and current · Paying the bad-tube event
- Section 6A first-domain route to full-wave continuity
Common forms and tangent graphs · The product regularity, and what it does not say
- Section 7A radial-interface parent with two retained flux sources
Finite source profiles really provide the required jets · Full continuity, the true core, and radial regularization
- Section 8Bare Coulomb with an unchanged Gaussian and prescribed fields
- Section 9An electron with a retained quantum oscillator source
- Section 10A distinct compact-entrance Coulomb alternative
- Section 11The quantitative cost of the anisotropic continuity route
- Section 12Why the complete current and original law matter
A hidden current that cancels after contraction · An absolutely continuous marginal does not admit a singular full law · A nonlocal current need not vanish at wave nodes · What a later record theorem must supply · Interfaces with preparation, records and coherent sources
- Section 13Conclusion
Funding, assistance and review
- Appendix AApproximate continuity and a whole-path source allowance
- OpeningBibliography