# Section 13: Conclusion

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## 13 Conclusion

<a id="p3r:conclusion"></a> A complete conservative quantum current can define deterministic reference-compatible histories under explicit density, action, logarithmic, Sobolev and boundary hypotheses. The reference density serves both as a compression measure and as the measure identifying exceptional entrances. It does not replace the original actual law. Once deterministic disintegration is established, every original absolutely continuous law is carried by the same measurable flow, while quantitative history estimates retain their original cap or truncation cost.

The stability result controls a supremum over the full history and separates two distinct tasks: approximating the complete density/current within a positive tube and paying for paths that leave it. Singular interactions and retained sources make those tasks substantive. The domain applications show how tangent regularity and first-domain control can contribute without assuming arbitrary powers of a Hamiltonian across a radial jump or Coulomb collision. Their quantitative constants, including the weak interpolation power in high dimension, remain part of the result's practical limitations.

A future physical application must independently justify one Hamiltonian and current constitution, its source and preparation law, the required estimates on the same parent, and a faithful record with a specified decoder and retention interval. These are distinct obligations; the flow theorem neither solves them by implication nor loses its mathematical content because they remain. The signed-source appendix provides a separate controlled comparison when an approximate reference is not exactly conservative.



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### Funding, assistance and review

 This research was conducted independently without external funding. AI systems assisted with drafting, mathematical checks, literature verification and manuscript preparation. The author remains responsible for the arguments and presentation; AI assistance does not constitute independent scientific validation. Collaboration on independent mathematical verification, computational replication or physical implementation is welcome.