Returning a quantum wavefunction and its Hamiltonian to their initial values need not return the configurations guided by that wavefunction. This paper uses the configuration changes left by exact quantum return cycles to characterize the Born probability law at a single preparation. For any prescribed finite connected particle graph, the manuscript constructs an engineered harmonic network with fixed, nonzero, directionally restricted pair interactions. Local one-body scalar controls implement exact nonlinear and Gaussian return protocols. The Born measure is proved to be the unique probability law invariant under the resulting library of configuration maps. The uniqueness theorem covers all Borel probabilities, including singular laws, without assuming a probability density or a regular assignment across wavefunctions. The paper supplies the guided-flow constructions and exact endpoint corrections, establishes a broader Gaussian holonomy result for connected quadratic networks, and develops consequences for reversible preparation transport and calibrated readouts. A randomized return controller is also analyzed. Absolutely continuous initial laws converge to equilibrium in total variation, but a quantitative obstruction prevents exact equilibration after finitely many rounds. Retained command history permits an inverse echo that recovers the initial nonequilibrium. Applying the uniqueness theorem to an actual preparation requires the explicit statistical premise of invariance under the specified returns. The manuscript presents self-contained mathematical results within its stated engineered model; it does not derive that premise or establish laboratory realization.
Returning a quantum wavefunction and its Hamiltonian to their initial values need not return the configurations guided by that wavefunction. This paper uses the configuration changes left by exact quantum return cycles to characterize the Born probability law at a single preparation. For any prescribed finite connected particle graph, the manuscript constructs an engineered harmonic network with fixed, nonzero, directionally restricted pair interactions. Local one-body scalar controls implement exact nonlinear and Gaussian return protocols. The Born measure is proved to be the unique probability law invariant under the resulting library of configuration maps. The uniqueness theorem covers all Borel probabilities, including singular laws, without assuming a probability density or a regular assignment across wavefunctions. The paper supplies the guided-flow constructions and exact endpoint corrections, establishes a broader Gaussian holonomy result for connected quadratic networks, and develops consequences for reversible preparation transport and calibrated readouts. A randomized return controller is also analyzed. Absolutely continuous initial laws converge to equilibrium in total variation, but a quantitative obstruction prevents exact equilibration after finitely many rounds. Retained command history permits an inverse echo that recovers the initial nonequilibrium. Applying the uniqueness theorem to an actual preparation requires the explicit statistical premise of invariance under the specified returns. The manuscript presents self-contained mathematical results within its stated engineered model; it does not derive that premise or establish laboratory realization.
What singles out the Born probability distribution in Bohmian mechanics when the same probability assignment must remain consistent across different physical controls? This paper proves a conditional uniqueness theorem for quantum equilibrium in nonrelativistic interacting systems. Under explicit regularity assumptions, a normalized probability assignment depending only on the current wavefunction must equal the Born distribution if it is consistent with Schrödinger evolution and Bohmian guidance across the admitted local scalar potentials. The pair interactions remain fixed, connect the particles, and have genuine mixed spatial dependence. The proof does not require additional controllable many-body potentials. The required controls can be reduced to translates and real amplitudes of a single suitable spatial profile for each particle. The manuscript also provides extensions to effective spin and identical-particle sectors, a separate characterization based on binary-flag heredity, and continuity conditions extending the results to wavefunctions with nodes. Consequences for apparatus records are stated under explicit preparation and measurement-coupling assumptions. The result characterizes the Born law within the stated statistical framework. It does not derive the control-consistency premise from deterministic dynamics or establish relaxation from arbitrary initial distributions. This revised preprint presents the theorem statements and proofs in a self-contained form for independent scholarly assessment.
Under what conditions does a quantum probability current determine complete, deterministic configuration histories, including nodes, singular interactions, and dynamical source coordinates? This paper establishes a theorem framework connecting quantum continuity equations to deterministic flows under explicit regularity, integrability, and boundary assumptions. The main result proves avoidance of wave nodes throughout a finite time interval and deterministic selection of reference-compatible paths. Uniqueness is established within a specified class of path laws whose time marginals are bounded by a finite multiple of the reference density. Any original configuration law absolutely continuous with respect to the initial wave density is then transported by reweighting those same paths, without introducing a new statistical population during evolution.The paper also proves approximation and stability results for entire histories. These require control of both the complete density and current, and explicitly account for trajectories leaving the region where the comparison estimates apply. Applications include discontinuous radial interactions with finite spin and quantum sources, bare Coulomb interactions, and a Coulomb electron coupled to a retained quantum oscillator. A separate theorem bounds endpoint and crossing errors for suitably regular comparison currents with a signed continuity defect. Counterexamples identify failures caused by omitted source currents, singular initial laws, and insufficient control near nodes. These results provide mathematical foundations for deterministic quantum transport and record analysis. They do not select an initial Born distribution or independently establish a complete measurement apparatus.
Can an autonomous quantum apparatus produce approximately Born-distributed outcomes and faithfully preserve its own earlier measurement label without assuming complete initial quantum equilibrium? This paper presents two finite autonomous Schrödinger–Bohmian constructions with explicit error bounds under restricted nonequilibrium initial laws. Each construction addresses three separate requirements: outcome calibration, copying the actual earlier configuration label, and retaining that label throughout a specified holding interval. The first model combines periodic focusing, a finite recoiling reference rotor, and a delayed compensated oscillator. Its record-distribution error is at most 0.004434840087, while failure to copy and retain its actual earlier label is at most 0.002327254798. The second model begins with a small radial Gaussian, includes autonomous preparation and activation, and incorporates the record writer from the original entrance. The joint law of its earlier label and complete symbolic holding record differs from an ideal constant Born-labelled record by less than 0.006086770113 in total variation. Its actual-label copy-and-hold failure probability is below 0.000552421956. The proofs combine bounded-variation calibration, exact recoil reduction, conditional-rank current estimates, finite-clock derivative bounds, and absolute probability-current estimates over the entire holding interval. The specialized derivations and coefficient calculations are included in the manuscript’s appendices. The two models retain distinct Hamiltonians and statistical assumptions. Neither establishes universal relaxation from arbitrary initial laws, supplies physical sources for every prescribed interaction, or demonstrates laboratory realization. External specialist verification remains open.
A measurement apparatus must do more than produce the correct final outcome frequencies: its records must faithfully preserve its own earlier declarations. This paper constructs autonomous quantum measurement chains that address both requirements within a specified nonrelativistic Schrödinger–Bohmian model. Assuming Bohmian guidance, complete initial quantum equilibrium, and a finite apparatus stock prepared independently of the unknown input, the manuscript gives a Hamiltonian bounded below that implements each fixed finite measurement programme to any prescribed positive tolerance. The complete model retains the source, pointers, internal clock, coherent resource products, copied archives, reset receivers, and an inaccessible quantum reference. Forced harmonic traps supply exact pointer motion. Retained orthogonal keys permit stationary record storage, while explicit copying and transfer preserve earlier records before working registers are reset. A finite-mass quantum clock replaces external time-dependent control. Weighted derivative estimates quantify its approximation error and bound the absolute probability current through archive decision surfaces. The resulting theorem separates final quantum-output accuracy from historical-record fidelity, including preservation throughout designated holding intervals. It retains coherent null outcomes and provides reproducible finite examples. Every positive tolerance uses finite resource choices for the fixed programme and observation horizon, although Gaussian packets retain unbounded tails. This is a conditional implementation and error-control result. It does not derive the initial Born law, uniquely determine Bohmian guidance, establish closeness of complete microscopic trajectories, or demonstrate laboratory realization of the admitted interactions.
This paper develops a semibounded material realization of a finite quantum measurement chain within a continuous-configuration theory. Its physical premises are spinor Schrödinger dynamics, Bohmian kinetic-momentum guidance and complete initial quantum equilibrium with a finite prepared resource stock. Smooth forced harmonic traps provide explicit pointer trajectories and crossing-time distributions. The construction retains pending excitation, loss products, fuel, reset receivers and inaccessible references. Physical archives, transported-trap reset and noncommuting continuation are incorporated into the same material inventory. A retained massive controller implements the programme autonomously. Weighted derivative estimates bound the complete retained-wave error, while absolute surface-current estimates separately control historical archive corruption. The closure theorem combines output, archive and pre-reset transfer errors. Explicit rival dynamics demonstrate that reliable records and Born-compatible endpoint statistics do not uniquely determine the underlying motion law. This is a separate constitutive completion, not a derivation of discrete Bell jumps. Guidance and initial equilibrium remain postulates, and the conclusions concern fixed finite programmes within the stated domain. This Version 2 preprint is a self-contained companion to the monograph and pilot-medium paper.
Can reversible quantum dynamics prepare a subsystem for measurement while retaining the physical archives of its earlier history? This paper provides an explicit mathematical construction and identifies the conditions under which such preparation succeeds. An exact, smooth split–copy–return protocol couples a Gaussian writer to a finite bank of position archives. The analysis shows why preparation of the writer alone is insufficient: singular archive populations and exact retained copies can preserve its fine information. A positive preparation theorem instead uses regularity and moment conditions on the complete original conditional law, retaining every archive coordinate and its correlations without imposing an upper bound on the probability density.A worked example with 102 position coordinates gives a joint conditional preparation error below 0.000010052. Under the stated assumptions, the same bound applies to a subsequent one-use labelled projective measurement with an arbitrary finite internal reference system. A separate physical receiver copies the writer’s earlier outcome and preserves it during the specified holding dynamics. The paper includes complete proofs, counterexamples, robustness estimates, and an analysis of reference systems and postselection. Its results concern conditional subsystem preparation and a one-use measurement within a controlled canonical-current model. Physical realization of the controls and independent justification of the admitted initial statistical laws remain separate requirements.