# Section 8: Discussion

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<a id="section-8"></a>

## 8 Discussion



The principal theorem identifies the Born density from an alternative statistical premise: one regular probability assignment must be compatible with a family of local controls in a fixed interacting system. The interaction supplies the connection between particle blocks through mixed coordinate conjugation. The functional consequences of this connection are much larger than the physical control list, which explains why a translated one-body profile can suffice for a uniqueness theorem without being a universal experimental actuator.

The result answers a precise version of the universal-equivariance question posed in [[6](/quantum-measurement/research/control-consistency/bibliography#bib-GS2007)]. Its conditional content is substantial: under the stated regularity and state-domain assumptions there is no competing density assignment. It does not derive those statistical assumptions from deterministic dynamics. In particular, allowing an extra preparation-history argument changes the question. The theorem is independent of exact nonlinear preparation-return constructions and requires no assumptions about their existence.

Spin, symmetry, and nodes require the hypotheses stated for them. Within those domains, the characterized equilibrium law has the familiar consequences for physical records. The mathematical contribution is the characterization and local-control reduction; the established Bohmian measurement analysis explains how that law appears in an experiment.