We formulate a regularized reciprocal extension of quantum hydrodynamics motivated by action-reaction symmetry in pilot-wave theory and by Sutherland-type post-quantum back-reaction schemes. The reciprocal anomaly is sourced by the imaginary part of the local momentum weak value. In the formal unregularized limit this gives the minimal closure ∆p proportional to Im(pw ). Since that reduced closure is singular at nodes and generically behaves as an effectively open dynamics, we construct a bounded completion on the torus by introducing: (i) a nodal regularization Aε =−(ℏ/2m)∇ln(ρ+ ε), (ii) a smooth information-limited mobility controlled by a nonnegative global admissibility functional built from relative entropy and regularized Fisher information, and (iii) a transported Eulerian memory field encoding cumulative reciprocal activity. The resulting model is a globally coupled nonlinear PDE for (ρ,S,M) with Hamiltonian advection, bounded reciprocal transport, capillarity-regularized quantum pressure, and local memory loading. We prove exact identities for weak-value decomposition, mass conservation, reciprocal-current entropy- gradient form, and memory-energy balance. We then state an energy-controlled local classical well-posedness theorem for fixed ε>0, based on a high-regularity Sobolev–capillarity energy. A modal asymptotic result shows that if the mobility exponent satisfies n>1, ultraviolet amplification is suppressed in a Fisher-dominated fixed-amplitude scaling regime, with an extended condition for amplitude-decaying spectra. In reciprocal equilibrium the model reduces to a regularized Madelung-type quantum hydrodynamics, recovering the standard unregularized system formally as ε →0 away from nodal singularities. The framework remains a candidate closure rather than an established physical theory, but its mathematical content is explicit enough to be analyzed, simulated, and falsified.
We formulate a regularized reciprocal extension of quantum hydrodynamics motivated by action-reaction symmetry in pilot-wave theory and by Sutherland-type post-quantum back-reaction schemes. The reciprocal anomaly is sourced by the imaginary part of the local momentum weak value. In the formal unregularized limit this gives the minimal closure ∆p proportional to Im(pw ). Since that reduced closure is singular at nodes and generically behaves as an effectively open dynamics, we construct a bounded completion on the torus by introducing: (i) a nodal regularization Aε =−(ℏ/2m)∇ln(ρ+ ε), (ii) a smooth information-limited mobility controlled by a nonnegative global admissibility functional built from relative entropy and regularized Fisher information, and (iii) a transported Eulerian memory field encoding cumulative reciprocal activity. The resulting model is a globally coupled nonlinear PDE for (ρ,S,M) with Hamiltonian advection, bounded reciprocal transport, capillarity-regularized quantum pressure, and local memory loading. We prove exact identities for weak-value decomposition, mass conservation, reciprocal-current entropy- gradient form, and memory-energy balance. We then state an energy-controlled local classical well-posedness theorem for fixed ε>0, based on a high-regularity Sobolev–capillarity energy. A modal asymptotic result shows that if the mobility exponent satisfies n>1, ultraviolet amplification is suppressed in a Fisher-dominated fixed-amplitude scaling regime, with an extended condition for amplitude-decaying spectra. In reciprocal equilibrium the model reduces to a regularized Madelung-type quantum hydrodynamics, recovering the standard unregularized system formally as ε →0 away from nodal singularities. The framework remains a candidate closure rather than an established physical theory, but its mathematical content is explicit enough to be analyzed, simulated, and falsified.
This paper folds a minimal full closure into a previously closure-neutral source-term framework for Sutherland-type action-reaction pilot-wave dynamics. The aim is not to claim an accepted extension of quantum mechanics, but to define a mathematically closed effective post-quantum candidate whose assumptions, conservation properties, and falsification conditions are explicit. The central object is the guidance mismatch field Delta p(x,t). In the closed model this mismatch is no longer free: it is fixed by a regularized weak-osmotic closure, Delta p = -lambda R_delta[Im(p_w)], where Im(p_w) is the imaginary component of the local momentum weak value. The raw Sutherland source s = (1/m) grad rho * Delta p is then globally compensated by sigma = s - Lambda rho, with Lambda chosen so that int sigma d^3x = 0. This produces local source-active redistribution without global norm creation. The corresponding wave equation contains a trace-preserving non-Hermitian effective term, i hbar (sigma/2rho) psi, while the particle-sector velocity is v_B + Delta p/m. The result is a closed dynamical loop psi -> rho,S -> Delta p -> s -> sigma -> psi, together with a particle-sector transport law. Downstream derivations are revised accordingly: branch drift becomes S_a - Lambda N_a, tunnelling corrections become S_B - Lambda N_B, local amplification requires s/rho - Lambda > div v_B, and linear onset is governed by deviation from the global source-per-mass average. The paper also preserves the frameworks epistemic boundary: no-signalling, relativistic extension, and derivation of lambda, delta, and the regularization length remain open problems.
We present a deterministic extension of de Broglie-Bohm pilot-wave theory that resolves the fundamental action-reaction asymmetry of the standard formulation. By introducing a reciprocal coupling term sourced by the imaginary part of the momentum weak value, Im(pw) = -(ħ/2)∇ ln ρ, we enable the particle configuration to influence the guiding wavefunction locally. To prevent the anti-diffusive instabilities inherent in such back-reaction models, we introduce a parameter-free "Fluctuation Clamp" mechanism. This mechanism budgets allowed gradient sharpening against a numerical vacuum surrogate scale, ensuring stability over 10^5 simulation steps while preserving time-symmetry and normalization. The model recovers standard quantum mechanics in equilibrium (ρ = |ψ|²) and in the weak-coupling limit (λ → 0), but predicts falsifiable deviations in non-equilibrium regimes, specifically in neutron interferometry (velocity shifts Δv ~ 5 × 10^-3 m/s) and tunneling time measurements (dwell time shifts δτ/τ ~ 0.1).
This companion appendix provides a standalone numerical supplement to the current reciprocal quantum hydrodynamics manuscript. The goal is narrow and explicit: to convert the paper’s one-dimensional periodic Weaver verification protocol into an executable, auditable simulation model. The supplement records the dimen- sionless verification system, the numerical scheme, the diagnostic contract, a baseline full-system run, a modal verdict sweep across the mobility exponents n= 0.5,1.0,1.5, and a nodal ϵ-descent stress test. The main numerical findings are that mass is preserved to machine precision in the reported runs, the high-mode modal response exhibits the predicted threshold ordering across the n= 1 transition, and decreasing ϵ strongly amplifies the anomaly magnitude near nodal structure while remaining controlled in the present regularized reduced solver. These results strengthen the manuscript’s verification and falsifiability claims, while leaving the singular ϵ→0 limit, higher-dimensional non-normality diagnostics, and experimental calibration as open targets.
A mathematically explicit extension of quantum hydrodynamics is developed in which a scalar coherence field ΦG modifies phase transport and is driven by the information-geometric structure of the probability density together with probability transport itself. The construction starts from the Madelung representation Ψ = √ρ eiS/ℏ, augments the phase gradient by the covariant shift ∇S ↦ ∇S − αℏ∇ΦG, and defines the dynamics through a variational (ρ, S) sector together with a dissipative Onsager-type relaxation law for ΦG. This yields a closed PDE system consisting of a continuity equation, a modified Hamilton–Jacobi equation, and a coherence-field equation that reduces in the overdamped limit to an elliptic closure. The source of ΦG is neither bare mass density nor bare probability density: it is a local Fisher-information density together with probability transport, so that ΦG measures coherence demand. The field equations are written explicitly; the coupling sign is fixed by an energy-stability argument, the transport-forcing sign by a dissipation argument, and the induced coherence-pressure term is computed exactly. A structural no-signalling condition is obtained by restricting the nonlinearity to Fisher-marginal source functionals that are blind to the entanglement Fisher excess under marginalisation. A coarse-grained Yukawa sector is shown to admit a conditional Newtonian limit under explicitly stated screening and additivity assumptions; this remains a conjectural macroscopic regime rather than a completed derivation of gravity. The paper closes with a falsifiable numerical programme—ablation studies, dispersion diagnostics, Fisher-information monitoring, and an explicit pilot solver scaffold—together with a claim-status ledger, a testability map, and a development roadmap that separate what is defined, what is derived, what is heuristic, and what remains unproven. Keywords: quantum hydrodynamics; Fisher–Rao geometry; Madelung dynamics; coherence field; no-signalling; information geometry.
Abstract Standard pilot-wave theory contains a structural asymmetry: the wave guides the particle, but the particle exerts no reciprocal influence on the wave. This paper develops a geometrically complete extension in which action–reaction symmetry is restored. Particle deviation from guidance is sourced by the imaginary part of the momentum weak value, a quantity intrinsic to quantum theory that vanishes in equilibrium and activates precisely where amplitude structure exists. Unregulated back-reaction generates deterministic instability through positive feedback between deviation and density gradients. Stability is restored through geometric phase accumulation along deviating trajectories, which feeds back into the wave’s phase structure and reshapes subsequent guidance. The resulting framework remains deterministic and time-symmetric and yields three independent falsifiable predictions: the weak-value source of deviation, trajectory-dependent phase reshaping, and the existence of a critical feedback strength separating unstable and bounded dynamical regimes.
Standard pilot-wave theory exhibits a fundamental dynamical asymmetry: the wavefunc- tion guides the particle trajectory through the Bohmian guidance equation, yet the par- ticle exerts no reciprocal influence on the wave. This lack of action–reaction symmetry is conceptually problematic if the wavefunction is regarded as physically real. This work develops Geometric Weak Value Dynamics (GWVD), a deterministic and time-symmetric extension of pilot-wave theory that restores reciprocal coupling between particle and wave. The key mechanism is the identification of the imaginary part of the quantum momentum weak value as a deviation source proportional to the score function ∇ln ϱ of the prob- ability density. Because the score function defines the Fisher information geometry of statistical manifolds, the resulting dynamics can be interpreted as a geometric flow on the manifold of probability distributions. Unregulated deviation produces anti-diffusive am- plification of density gradients, corresponding to runaway growth of Fisher information. Stability is achieved through trajectory-dependent geometric phase accumulation, which feeds back locally into the wavefunction phase and introduces curvature-dependent reg- ulation of particle motion. Crucially, a Lagrangian formulation is presented from which both the modified Schrödinger equation and the modified guidance equation are derived simultaneously via a single variational principle, establishing equivariance by construc- tion and reducing the free parameters of the theory to a single consistency condition. The theory predicts three experimentally testable signatures: deviation velocity proportional to Im(𝑝𝑤), trajectory-dependent interference modulation via geometric phase, and a criti- cal feedback threshold 𝛼crit = λ/2 separating unstable and bounded regimes. Extension to higher spatial dimensions reveals emergent vorticity, vortex lattices, and mode-selective pattern formation. GWVD maps onto a minimal Turing activator–inhibitor system, suggest- ing a deep connection between quantum dynamics, information geometry, and nonlinear pattern formation.