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.
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.
How can an approximation to a quantum wave be converted into a reliable estimate of physical transport when coherent sources, singular interactions, and continuous spectra remain part of the dynamics? This paper develops quantitative response and probability-current estimates that retain the complete quantum source, interference terms, and reciprocal feedback. The results connect temporally regular forcing to response bounds that control the derivatives needed for physical currents. They include finite spectral-window estimates for varying coherent inputs, ordered resolvent identities for noncommuting energy weights, and complementary spatial-force and temporal-regularity methods. Bound-state poles and continuum contributions are retained explicitly, together with endpoint, localization, and source-regularity costs. A worked application treats a Coulomb electron coupled to a dynamical quantum oscillator through a smooth compact interaction profile. It derives the complete Coulomb forcing measure and establishes quantitative spectral and finite-time estimates for the coupled model. The analysis controls both electron and source currents without replacing the source by a classical expectation or discarding its feedback. Appendices develop forcing compression with all sources retained and demonstrate why a small omitted source contribution need not imply negligible causal feedback. These results supply explicit analytic inputs for quantum transport and measurement-record analysis. The current bounds are not themselves event probabilities. Such probabilities, and a physically realized measurement apparatus, require additional flow, preparation, and decoder assumptions.
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.
Discrete measurement records coexist with continuously variable phase, orientation and reference-frame comparisons. We call coherent admissibility of alternatives the probability critical line (PCL), and probability conditioned on a known record the probabilistic line of classical probability (PLP). Each newly known record re-anchors the remaining comparison interval; later predictions retain the full history and measurement backaction. We establish two conditional results. A detector-relative sign is stable when its comparison margin exceeds the probe error; its normalised geometric ambiguity scales inversely with the resolved comparison length at fixed additive error. An explicit additional-coordinate rule assigns one record at every step of a finite measurement history and preserves all adopted conditional quantum probabilities. The rule retains linear global quantum evolution and rejects wavefunction completeness as an assumption of the proposed account. Coherent recombination, passive frame invariance, sequential conditioning and finite-resolution bounds are verified numerically using specified synthetic inputs. Existing electron interference, single-spin readout and coherent-spin-control experiments constrain the interpretation. Geometric rest, internal evolution and photon signal delay have separate definitions. The resulting closure is a consistency construction for detector-relative records. Its identification with a physical polar-contact mechanism requires a coupling law that produces the selected apparatus record. Ai genetated content and interpretation may not contain exact source logic chain and reasoning independently established by the Author. Original source material has been timestamped for proof of prior art.
We construct a source-matched reference calibration for distinguishing the coherent law h = 0 from the specified added generator h = J/25 in a finite quantum-record model. A complete-carrier transfer bound connects an accessible reference contrast to a delayed target contrast while retaining the original drift and actual law. On reference coverage and measured admission, restricting both hypotheses by the same observed reference band yields a directed target-set separation of at least 0.0071230619 whenever both compatible classes are nonempty. Rival worlds may have different admitted nuisance parameters. The constants are uniform over independently known finite lengths m >= 65, with tag 65, access prefix 1:32 and target deadline 25.02/J. Finite reference preparation and work/source reset satisfy the declared response budgets. Reset acts on the actual joint input and consumed supply, with the retained reference R exactly inert. It does not assign a universal reduced channel to correlated supplies. Under almost-sure conditional resource and supply bounds, the comparison extends to history-dependent acquisition. A prespecified rule, including abstention below the calibration threshold, has each-law unconditional wrong-declaration probability at most 2 exp(-6) + exp(-85/16) < 0.01. This is neither a posterior probability nor an error bound conditional on admission. The statistical step uses standard bounded-martingale concentration. Global target states and specified auxiliary supplies remain resources. Reset requires additional globally conditioned controls and has very large finite costs. The result is a conditional theorem for the two declared laws; it does not establish practical implementation or a unique diagnosis of fundamental collapse. A supporting Lean 4 companion contains 85 theorem declarations and 30 supporting definitions across Calibration, Operations and Risk. It provides checked scalar, sufficient-bound, resource-arithmetic and finite-history risk consequences under explicit premises. The companion has an executed Linux preflight and accepted returned macOS compilation, named-type and transitive-axiom evidence. The quantum constructions and conditional moment, filtration, tower and tail derivations remain written proofs or external premises; this is not an end-to-end formalization of the physical theorem chain. The original checked companion targeted author v0.4 and is carried forward unchanged, with its scope and execution chronology preserved.
How can deterministic microscopic dynamics reproduce quantum jump histories, including their timing, while creating records faithful to actual earlier events? This paper constructs a conditional realization of the minimal Bell jump process using a deterministic hybrid model on a fixed finite configuration graph. The model combines a canonical complex field, conservative packet production, carrier–packet reactions, finite-rate recombination, and a finite spatial gas. Under explicit initial calibration and gas-independence assumptions, the ordinary-configuration path law converges in total variation to the Bell jump process without rescaling physical time. Quantitative estimates control the gas contact history, recombination error, and population tracking, including wavefunction nodes, current reversals, and zero-current intervals. An autonomous quantum clock carries a finite measurement circuit that retains fuel, loss channels, copied archives, reset receivers, feedback, and an inaccessible reference. At specified copying gates, the Bell path crosses the gate exactly once during the first pass. The archive therefore records its own actual earlier configuration and preserves that record under subsequent label-preserving operations. An explicit resource hierarchy provides an input-uniform polynomial convergence bound for a fixed engineered circuit. The theorem depends on the stated interaction catalogue and statistical preparation premises. It does not derive those premises, establish laboratory realization, or embed the complete hybrid dynamics in a single smooth material Hamiltonian. A supplementary smooth construction covers isolated contact modules.