Discussion

9 Discussion

The exact return library constrains a probability law through the configuration history left behind by quantum cycles. Its nonlinear plane returns determine a marginal among all Borel probabilities, and its Gaussian returns rotate every configuration direction. The combination selects the Born law without a regular density functional across wavefunctions and without excluding singular candidate measures.

The physical restrictions are constructive parts of the result. Pair interactions are nonzero and fixed on a connected prescribed graph, but they are directionally engineered to leave a holding plane available. The local quadratic controls are unrestricted finite trap matrices; nonlinear controls are one-body scalar potentials with the growth and regularity proved above. The exactness statements use finite-dimensional endpoint correction and exact density histories, not approximate recurrence or finite oscillator cutoffs.

The Gaussian extension in section A removes the weak coupling and special spectrum of the original Gaussian device. It does not remove the structural restriction in the nonlinear uniqueness theorem. Obtaining a nonlinear exact return with sufficient radial action in a fully coupled isotropic network remains a separate extension.

Finally, invariant-measure uniqueness does not establish a relaxation rate or convergence from every initial probability. It also does not establish that an actual preparation has the invariance property. With that property, the present theorem identifies the equilibrium law; standard modeled readouts then give its record statistics. The statistical premise and the dynamical construction have distinct physical content.