Introduction

1 Introduction

Publication relationship.

This article revises the author's manuscript of 7 September 2026. Its nonlinear and Gaussian constructions, all-Borel uniqueness theorem, general Gaussian appendix, and quantitative witness are retained with full proofs. Section 8 consolidates the same-date randomized-return preparation report [18]; these results are not claimed as newly discovered in this revision. The companion common-control density characterization is logically independent and is cited only for comparison.

Suppose a preparation is taken through a controlled Schrödinger evolution and brought back to precisely the same wavefunction ray and Hamiltonian. Must its configuration distribution also return? In Bohmian mechanics the question is substantive: the velocity depends on the whole wavefunction history, and the endpoint transformation of configurations need not be the identity. We call this transformation a preparation-return map. The collection of such maps can distinguish probability laws even when each quantum endpoint is identical.

We construct an interacting model in which invariance under a specified library of exact returns singles out the Born measure among all Borel probabilities. The argument has two ingredients. Nonlinear density loops in a reserved two-dimensional plane determine its marginal law, including possible singular parts. Gaussian loops mix all configuration directions. Rotational invariance and one Gaussian marginal then determine the entire joint distribution by its characteristic function. This last step requires neither independent coordinates nor conditional densities.

The distinction between the three assertions involved is important. The Hamiltonians implement exact returns; their configuration maps have a unique invariant probability; and an actual preparation law may be required to be invariant under those returns. The first two assertions are proved here. The third is a statistical preparation principle, not a consequence of deterministic Schrödinger and guidance equations.

1.1 Main result and scope

Particles are distinguishable, scalar, and three-dimensional. We set $\hbar$ and their masses to one. A separate positive mass for each particle is accommodated by blockwise mass scaling. Let $G$ be a prescribed connected graph on $N\geq2$ particles, and write $d=3N$.

Theorem 1.1 (Engineered return uniqueness)

For every such graph there exist fixed nonzero pair potentials

$$W_{ij}(q_i,q_j)=\frac{k_{ij}}2(q_i^1-q_j^1)^2,\qquad k_{ij}>0,\quad \{i,j\}\in G,$$

and one-body harmonic holding traps with positive total stiffness $K_0$, having the following properties. The coordinates $s=(q_1^2,q_1^3)$ form an independent harmonic plane at holding, with frequencies $1,2$. Put $A_0=K_0^{1/2}$ and

$$\psi_0(q)=\left(\frac{\det A_0}{\pi^d}\right)^{1/4} e^{-q^TA_0q/2},\qquad \nu_0(dq)=|\psi_0(q)|^2\,dq .$$(1.1)

There is a library $\calH_{\psi_0}$ of globally defined smooth configuration diffeomorphisms, generated by the exact protocols specified in section 3, section 4, section 5, such that

$$F_*\mu=\mu\ \text{for every }F\in\calH_{\psi_0} \quad\Longleftrightarrow\quad \mu=\nu_0$$(1.2)

for every Borel probability $\mu$ on $\R^d$.

Each protocol has finite duration, leaves every pair potential fixed, uses only one-body scalar controls, and satisfies $\psi(T)=e^{i\theta}\psi_0$ and $H(T)=H(0)=H_0$ exactly. The controls are local quadratic traps and smooth nonlinear potentials on the reserved plane. The latter differ from its harmonic holding potential by at most linear growth. All guided histories used in the theorem exist at every configuration.

The quantifier is existence of engineered couplings on each graph, not universality over its coupling matrices. Their interparticle blocks have rank one and leave the reserved plane uncoupled at holding. Local quadratic stages subsequently mix that plane with the rest of the network. The holding ground state is entangled across interacting particle partitions, although its reserved-plane factor separates. The proof does not assign a factorized law to an arbitrary candidate $\mu$. Full isotropic pair interactions generally remove this plane; the nonlinear theorem makes no assertion for them.

1.2 Relation to previous work

Quantum equilibrium, effective wavefunctions, and the extraction of measurement statistics from actual apparatus configurations are established parts of Bohmian mechanics [3, 4]. Goldstein and Struyve [7] prove uniqueness under a locality hypothesis on a wavefunction-indexed probability functional and discuss other possible characterizations, including universal equivariance and heredity. Our object is instead a single arbitrary Borel law at one preparation. The companion control-consistency characterization [19] concerns a differentiable probability assignment across states and is not used in any proof below. The respective assumptions are different; we assert no implication ordering them.

Two transport results identify relevant geometric mechanisms. Abdelgalil and Georgiou's Gaussian transport theorem [1, version 2, Proposition 4 and Corollary 1] gives the full covariance-preserving rotation group for unrestricted Gaussian horizontal transport. Their smooth transport analysis [2, version 2, Proposition 1, equation (10)] provides the gradient-bracket identity used below. We reproduce that identity and its local consequence. Our problem additionally restricts the physical actuators and requires exact Schrödinger return. In particular, a covariance path by itself does not establish its realizability by fixed-interaction one-body controls.

Recurrence and control of quadratic oscillators were studied by Genoni et al. [6]. Recurrence permits approximate reversal of positive quadratic dynamics. Here an endpoint submersion supplies exact corrections, and a separate calculation determines their Bohmian configuration action. Inverse engineering from flow fields has precedents in shortcut-to-adiabaticity constructions [9]. Our nonlinear argument supplies a weighted Poisson inverse and global estimates on Gaussian tails. The semigroup method is closely related to Stein-factor estimates for strongly log-concave densities [8]; the parameter and all-order spatial estimates needed here are proved directly.

The contribution is this physical realization and invariant-measure argument in a specified interacting network, rather than a new derivation of the underlying Bohmian measurement framework.