Admissible Source-Active Quantum Support: A Two-Boundary Trace-Preserving Closure for Transport Defect Geometry, Weak-Osmotic Reaction, and Configuration-Space Connectivity
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.
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.
This paper develops a mathematical framework for two-boundary quantum admissibility. Given an initial wave field ψi = √ρiexp(iSi/ℏ) and a final, post-selected, or backward-evolving effect wave field ϕf= √ρf exp(iSf/ℏ), the central question is not merely where both amplitudes are non-zero. The question is whether an intermediate support structure can simultaneously satisfy the transport requirements imposed by both boundaries. The framework introduces a log-transport defect Dα associated with the power-family density ρα = ρα i ρ1−α f transported by the symmetric boundary velocity v= (ui+uf)/2. The defect admits an exact decomposition into an irreducible geometric obstruction, a density-reweighting channel, and one-boundary continuity residuals. This decomposition generates a hierarchy of repair classes ranging from global density reweighting through local density repair and finally to topological obstruction characterized by nodal winding mismatch. The framework lifts naturally from ordinary space to full configuration space, where the primary object is not a trajectory but a connected component of low-obstruction support. These components, termed admissible support tubes, provide a geometric description of conditional event connectivity under two-boundary constraints. The framework does not modify the Born rule, does not introduce hidden signalling, and does not identify weak values with literal particle trajectories. It proposes a diagnostic geometry for conditional quantum support based on exact transport compatibilit
This document develops a mathematical framework for two-boundary quantum admissibility. Given initial and final wave fields ψi = √ρieiSi/ℏ and ϕf= √ρfeiSf /ℏ , one asks when an intermediate local support structure is compatible with both boundary data. The central diagnostic is the log-transport defect of a power-family density ρα = ρα i ρ1−α f transported by the symmetrized velocity v = (ui + uf)/2. 1This defect decomposes exactly into an irreducible geometric obstruction, a density- reweighting channel, and one-boundary continuity residuals. The scalar parameter α is thereby recognized not as the whole theory, but as the first repair test. Scalar reweighting removes only a one-dimensional projection of the geometric-mean defect. Promoting α to a local field α(x,t) converts the repair problem into a first-order transport equation, and failure of that equation gives a stronger form of obstruction. Finally, nodal winding mismatch TC = m 2πℏ C(ui−uf)·dℓ= ni(C)−nf(C) is invariant under scalar and local density reweightings, yielding a non-reweightable topological obstruction. The framework lifts naturally to configuration space, where physical event succession is described by admissible support connectivity rather than by ordinary- space path continuity. Classical trajectories arise as narrow projected support tubes, while quantum nonclassicality appears when admissible connectivity persists in configuration space although its ordinary-space projection is broad, thresholded, or coarse-grained. Coherence dynamics and holonomy sectors are discussed as extensions of the admissibility hierarchy rather than as part of its theorem-level core.
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.
This paper develops a geometry-first account of quantum transport in which admis- sible local flow is constrained by the nodal and phase structure of a two-boundary wave field. The central quantitative device is the power-family density˜ ρα ∝ρα i ρ1−α f together with an obstruction decomposition of the numerically computed log-transport defect into an irreducible geometric term A, a density-reweighting term B, and weighted one- boundary correction fields. Two benchmark models were examined: a driven anisotropic harmonic packet mismatch (Model A) and a double-slit barrier with a postselected detector packet (Model B). In Model A the representative density scan has a sharp optimum at α= 0.2, reducing max Enat from 3.377 ×10−1 at α= 1/2 to 1.740 ×10−1 . In Model B the optimum is shallow near α= 0.5 and the minimum remains high, with max Enat = 6.516 ×10−1. Across both models, the full field (A+ B) + αΞi+ (1−α)Ξf reconstructs the computed defect to machine precision, with peak relative residuals of order 10−16, while the obstruction-only comparison remains O(1). The numerical evidence therefore distinguishes negotiable smooth mismatch from hard obstruction within the tested transport representation and supports a sharpened admissibility frame- work. A reservoir-like interpretation is presented only as a conjectural extension: the established result is the exact obstruction decomposition and its contrasting behaviour across smooth and interference-dominated geometries.
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.
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.