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Shadow Theory

Why the Born density?

Control Consistency and Born Equilibrium: Uniqueness from Local Potentials and Fixed Interactions

Local controls and fixed connected interactions select the Born density from a regular, control-consistent probability assignment.

Jeremy Rodgers · Independent Researcher · 4 October 2026

Revised preprint candidate

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15 reading sections16 statements14 proofs65 mathematical displays

The complete paper

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Every section, proof and appendix, with linked equations and the complete bibliography.

  1. OpeningAbstract and publication identity
  2. Section 1Introduction

    Publication relationship.

  3. Section 2Physical model and statistical assumptions

    Hamiltonians and a common smooth domain · Assignments, regularity, and consistency

  4. Section 3From phases to transport and uniqueness

    Statistical annihilators

  5. Section 4Local controls and fixed interactions
  6. Section 5Translated profiles and an interacting example
  7. Section 6Internal degrees of freedom
  8. Section 7Consequences for physical records

    Joint preparation and existence of trajectories · Implemented effects and instruments

  9. Section 8Discussion
  10. Appendix ANonzero approximation and nodes
  11. Appendix BA permutation-preserving refinement
  12. Appendix CAn alternative characterization from binary-flag heredity
  13. Appendix DA finite-time Gaussian pointer
  14. OpeningDeclarations

    Funding. · AI assistance and responsibility. · Independent review and collaboration. · Proof and verification materials.

  15. OpeningBibliography