This manuscript gives a strengthened first-principles closure of a weak-osmotic action-reaction pilot-wave dynamics. The aim is not to claim a final field theory, but to close the nonrelativistic finite-resolution source-active dynamics on a declared mathematical domain. The primitive wave ontology is , so that the local momentum weak value has the standard decomposition ; the imaginary part is precisely Nelson's osmotic velocity , well known from stochastic quantum mechanics. The saturation length is physically identified with the thermal de Broglie wavelength . The raw source-per-density field is projected in onto the additive subspace generated by operationally protected marginals; for any finite protected family and this subspace is closed and the projection is well defined. The empirical reaction current is the closest current to the bare weak-osmotic current subject to exact Born equivariance . The finite- resolution theory is locally well-posed. Continuum well-posedness, relativistic hypersurface integrability, and empirical calibration remain stated maturation tasks. A concrete bipartite benchmark density is specified in Appendix B for independent numerical reproduction.
This manuscript gives a strengthened first-principles closure of a weak-osmotic action-reaction pilot-wave dynamics. The aim is not to claim a final field theory, but to close the nonrelativistic finite-resolution source-active dynamics on a declared mathematical domain. The primitive wave ontology is , so that the local momentum weak value has the standard decomposition ; the imaginary part is precisely Nelson's osmotic velocity , well known from stochastic quantum mechanics. The saturation length is physically identified with the thermal de Broglie wavelength . The raw source-per-density field is projected in onto the additive subspace generated by operationally protected marginals; for any finite protected family and this subspace is closed and the projection is well defined. The empirical reaction current is the closest current to the bare weak-osmotic current subject to exact Born equivariance . The finite- resolution theory is locally well-posed. Continuum well-posedness, relativistic hypersurface integrability, and empirical calibration remain stated maturation tasks. A concrete bipartite benchmark density is specified in Appendix B for independent numerical reproduction.
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.
Thismanuscriptproposesastrongerunifiedtheorysynthesizingtwopreviouslyseparatedideas: two-boundary admissibility geometry and weak-osmotic source-active closure in Sutherland-type action-reaction dynamics. The first ingredient supplies a diagnostic geometry of compatibility for pre-/post-selected or initial/final quantum boundary data through a log-transport defect, density repair hierarchy, nodal winding obstruction, and configuration-space support tubes. The second ingredient supplies an effective closed reaction law in which a guidance-mismatch field is no longer arbitrary but is tied to a regularized imaginary weak momentum, producing a compensated local source that redistributes wave intensity while preserving total trace. The present synthesis argues that these structures should not remain independent. If two-boundary transport mismatch is physically meaningful, then the natural next step is not merely to diagnose where support is admissible, but to formulate a candidate compensated dynamics on admissible support and to state explicitly where density-only reaction must fail due to phase-topological obstruction. The resulting proposal is a two-boundary trace-preserving closure on an interpolating density family ρα = ρα i ρ1−α f and symmetric transport field v= (ui + uf)/2. A regularized weak-osmotic mismatch ∆pα is built from ∇ln ρα,δ, but its action is gated by an admissibility weight derived from the defect hierarchy. The effective source is written in exact replicator-compensation form σα = ρα(gα−⟨gα⟩ρα ), guaranteeing exact global norm preservation on the chosen regular domain. This makes it possible to distinguish kinematically repairable mismatch, dynamically amplifiable low-obstruction support, and topologically forbidden sectors that no density reweighting can resolve. Branch drift, barrier correction, tube reinforcement, and relative support selection are then reformulated as compensated source-per-mass imbalances on connected components of low-defect configuration-space support. The manuscript is explicit about status boundaries. The geometric defect decomposition, support-tube language, and topological obstruction analysis are exact at the stated regularity level. The trace-preserving source closure is a proposed effective extension, not accepted quantum mechanics. The no-signalling boundary is isolated as an explicit marginal-source condition and verified in the stated symmetry and barrier-gated sectors; a fully relativistic completion remains a separate extension. What is claimed is narrower but more useful: the combined framework yields a mathematically sharper theory of admissible source-active quantum support in which compatibility geometry and reaction dynamics constrain one another rather than remaining disconnected.
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 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 paper asks a deliberately narrower question than its predecessor: when does a Modal Triplet Theory (MTT) description reconstruct the standard nonrelativistic de Broglie-Bohm dynamics? We first specify the complete lower pilot-wave record. In addition to a complex Hilbert space and self-adjoint Schrodinger Hamiltonian, that record contains a physical configuration manifold, an actual configuration, a selected probability current, a guidance field and maximal flow, an equilibrium measure, subsystem and instrument data, a locality contract, and source and error certificates. For a regular spinless scalar Schrodinger model the standard current obeys a continuity equation, and velocity equals current divided by density away from nodes. Equivariance then propagates an initially chosen wavefunction-squared distribution. These facts are standard lower-level mathematics: continuity alone neither selects the actual configuration nor derives the initial equilibrium distribution, and it does not uniquely select the current because divergence-free modifications preserve the same continuity equation. The MTT result is therefore conditional. If one selected upper state emits the full pilot-wave record and its state, dynamics, configuration decoder, current, instruments, and records form commuting descent diagrams, then MTT reconstructs Bohmian trajectories and their operational statistics on the declared interval. The current canonical geometry is a ten-dimensional bundle over a four-dimensional Lorentzian base with a compact six-dimensional fibre; the former literal product of four-dimensional spacetime and three independent internal factors is not used. Failure of the reconstruction means that a required current, decoder, flow, or stable continuation is unavailable. It does not follow from the absence of a right inverse, and it does not establish that Bohmian mechanics is globally impossible. Standard global-existence results cover broad Hamiltonian classes for almost every equilibrium initial configuration. Entangled guidance is explicitly nonlocal in physical configuration space. Operational no-signalling can nevertheless hold in quantum equilibrium, while an MTT claim of upper-world locality requires a separate descent theorem preserving settings, local algebras, records, and independence assumptions. Measurement is treated as an ordinary system-apparatus interaction with an effective conditional wavefunction, not automatically as an admissibility-barrier crossing. The selected q=79 binary recorder provides an exact stopped operational output measure on its declared commuting Fock domain, but it does not yet source arbitrary Bohmian equilibrium ensembles or one uniquely actual history. The paper thus establishes a precise regime-limited reconstruction and a finite completion contract, not a universal derivation or a no-hidden-variable theorem.
We establish a logically closed, mathematically formulated, computationally executable, and experimentally falsifiable theory of wave dynamics within the global-realist framework. The construction inherits exactly three physical axioms: microscopic ontic definiteness and finite localization, the physical reality of the spacetime–vacuum substrate, and persistent causal source–response coupling. A physical wave is defined as a propagating or mode-transferring retarded disturbance generated by an actual finite source history, rather than as an independent substance or a universally ontic wave function. A unified causal chain is developed from finite-core source functionals and local, nonlocal, or hereditary medium response to constraint-compatible retarded Green operators, physical modes, energy–momentum and wave-action transport, scattering, interference, diffraction, nonlinear conversion, stochastic propagation, source reaction, and detector records. The theory treats strong, weak, and distributional solutions; hyperbolic, parabolic, dispersive, and mixed operators; gauge constraints; direct and tail propagation; spectral poles, branch cuts, continua, quasinormal modes, bound states in the continuum, and non-Hermitian representations. Controlled reductions include WKB rays, diffusion limits, transport equations, envelope models, solitons, shock laws, and classical branch theories with explicit error bounds and limit-order requirements. Scalar, electromagnetic, gravitational, acoustic, elastic, surface, internal, plasma, lattice, thermal, nuclear, matter, active, nonreciprocal, chiral, gyrotropic, and time-varying waves are internalized using unified notation while retaining their distinct sources, constraints, polarizations, constitutive structures, boundaries, nonlinearities, and observables. Executable regression suites verify smooth propagation, entropy shocks, passive memory, constraint preservation, dispersive transport, matched layers, resonance poles, quasi-BIC opening, spectral-pollution rejection, and an infinite-domain resonance computed independently by exact Dirichlet-to-Neumann reduction and exterior complex scaling. The resulting architecture distinguishes structural derivation, numerical verification, parameter calibration, and held-out experimental validation, thereby providing explicit rejection criteria for every admitted wave model. **Keywords** Global realism; wave dynamics; causal Green functions; finite-core sources; historical memory; wave propagation; dispersion; wave action; scattering; nonlinear waves; stochastic media; quasinormal modes; bound states in the continuum; topological waves; numerical verification; experimental falsifiability.