Control the current of the whole system
Complete-Current Estimates for Coherent Quantum Sources and Continua
Quantitative response estimates retain source motion, coherent interference, spectral poles, continua and feedback in the complete physical current.
Jeremy Rodgers · Independent Researcher · 9 October 2026
9 October 2026 manuscript
The argument
Keep the source, the interference, and the current
A controlled wave approximation becomes a transport estimate when it also controls the physical derivatives and every retained source current. This paper develops that conversion for coherent quantum dynamics, including singular interactions, bound-state poles, continuous spectra, and feedback from a dynamical source.
Several complementary routes supply the needed derivative control. Temporally regular forcing can be lifted into a physical form estimate; finite spectral windows can handle an input whose direction changes in Hilbert space; and ordered resolvents carry physical energy weights through an interacting parent. The order of the operators, source derivatives, and endpoint terms remains visible in each proof.
A Coulomb electron coupled to a quantum oscillator through a smooth compact interaction profile provides a worked application. The calculation retains the full bound-and-continuum forcing measure, the actual source interaction, and reciprocal feedback. A separate finite-time calculation bounds the integrated electron and source current differences by 0.046 and 0.004 in their respective coordinate lengths. Those bounds supply quantitative inputs for subsequent transport and record analysis.
Start with the physical current
The exact current formula includes covariant kinetic terms, coherent cross terms, and first-order drift. It identifies the wave and form norms every later response estimate must provide.
Read the argument →Use temporal regularity
An ordered inverse lift converts suitable forcing in the form dual into a first-form response. The common-form work bound keeps the time-dependent evolution under control.
Read the argument →Retain the bound states and continuum
The Coulomb calculation gives the bound atoms, continuum density, normalization, and sum rules for dipole forcing. These feed the quantitative estimates without dropping the spectral poles.
Read the argument →Restore the coupled source
The compact interaction retains its smooth collar, true operator and form domains, source coordinate, and actual coupled resolvent. The low-window estimate includes the source weight in the stated order.
Read the argument →Reach the finite-time currents
Source graph bounds and the feedback-modified input control both electron and oscillator currents over the specified horizon. The final estimates retain their physical coordinate units.
Read the argument →P4 supplies analytic current bounds for complete coherent parents. P3 provides compatible flow results for the retained electron–oscillator model and separate comparison theorems with their own regularity and tube assumptions. A record application must combine the relevant current estimate with its physical decoder, boundary geometry, retention interval, and original-law factor; P1 and P2 give explicit examples of preparation and record tasks.
The complete paper
Follow the full argument.
Every section, proof and appendix, with linked equations and the complete bibliography.
- OpeningAbstract and publication identity
- Section 1Introduction
Relation to established methods and prior work · Conventions and the structure of the results
- Section 2From a coherent error to the complete physical current
Coordinates, moving tests and the original law
- Section 3Temporal dressing in the physical form dual
Two sharp boundaries of the temporal argument
- Section 4A finite spectral window with a varying coherent input
Unbounded form coupling: near response and far dressing
- Section 5Ordered resolvents for an unbounded closed-form interaction
- Section 6Damped temporal response with a noncommuting physical form
- Section 7Whole-form spatial and source derivatives
- Section 8Spatial force and temporal regularity as distinct current resources
A force estimate with covariant drift · A positive invariant when a quadratic trap is inverted · Temporal transfer with a retained internal clock · Singular residuals and a safe common-domain alternative
- Section 9A complete Coulomb forcing measure
A continuum-only coherent response
- Section 10The actual compact-profile electron–source parent
Exact blocks, including source feedback · The spectral supremum and its correction
- Section 11Actual source graphs and finite-time complete currents
All coherent terms in the original physical currents · Flow and companion-paper interfaces
- Section 12Significance and remaining scope
Funding. · AI assistance and verification. · Independent review and collaboration.
- Appendix ASpatial forcing compression with all sources retained
- Appendix BA small omitted source does not remove causal feedback
- OpeningBibliography