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 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 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 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.
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 studies the fixed-background many-body extension and the Wallstrom obstruction within the scalar density-phase framework of capacity geometry. Within the same local first-order homogeneous density-phase action class as the general scalar forcing theorem, the forcing argument extends immediately to declared configuration-space backgrounds and yields the standard spinless many-body Schrödinger equation without any factorisation ansatz. The bare density-phase equations do not force Wallstrom quantisation: a punctured-disc family gives explicit finite-action counterexamples, and the sharpened locallocking relation reduces that family only continuously, not integrally. The strongest continuum closure theorem is a domain-selection statement: on the intrinsic nonvanishing complex-field domain the forced action is the Schrödinger action, Wallstrom counterexamples are excluded, and circulation quantisation is automatic by degree theory. What remains open is whether the SGOC construction selects that single-valued complex domain rather than only the larger density-phase representation. A second contribution is a precise theorem map for the discrete carrier-transport approach. Native phase-sensitive SGOC data live on the Fourier/triad skeleton, while the discrete Wallstrom problem needs carrier data on a spatial triangulation of a nodal-complement component. The missing theorem is a genuine native-to-carrier transport theorem. On closed orientable surface carriers the remaining cases split into three transport regimes. Once a global carrier edge-phase field exists, the total-curvature scalar obstruction vanishes automatically by simplicial Stokes; once non-flat support is confined to a proper carrier,homology-killing enlargement and relative H^1 complement globalisation handle the rest of the downstream topology. Selector-side reductions are also consolidated. Before localisation, the remaining selector-side existence content is one universal H^2 transgression class. After localisation to a proper carrier, existence of a strong completion is automatic, and the only surviving noncanonical part is the ambient surface period torsor — an affine A^{2g}-torsor on genus g — with complement residues canonically separated off. Exchange sectors and spin are additional topological or multicomponent input, not outputs of the present scalar variational route. Keywords: - Schrödinger equation - Capacity geometry - Wallstrom obstruction - Many-body quantum mechanics - Madelung equations - Circulation quantisation - Domain selection - Carrier transport - SGOC programme Related identifiers (references cited in this paper): - 10.5281/zenodo.19156067 (cites) — Capacity Geometry: REG, Dirichlet Tension, and Quasi-Local Mass (SGOC Volume 2, Part 1) - 10.5281/zenodo.19156316 (cites) — The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge (Part 1A) - 10.5281/zenodo.19189901 (cites) — The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background - 10.5281/zenodo.19259771 (cites) — Radial Surcharge Inversion and Free-Boundary Host Selection - 10.5281/zenodo.19267944 (cites) — Free-Boundary Surface Selection for a First-Order Nonlinear Profile Equation in Three Dimensions Related identifiers (external references): - 10.1103/PhysRevA.49.1613 (cites) — Wallstrom, T. C., "Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations," Physical Review A 49(3) (1994), 1613–1617
The completed Chronoflux Atlas recovers the quantum regime as the coherent phase branch of conserved temporal transport. Continuity supplies the current, temporal order and positive kinetic response recovered in DPL-00 through DPL-04. Coherence adds a non-vanishing complex state with a physical phase and action scale. Once that descendant is admitted, variation returnsphase transport, topology quantises circulation, the Fisher-gradient term returns the Bohm quantum potential, and the density-phase equations recombine exactly into the Schrödinger equation. Quantum behaviour is therefore not attached to continuity by analogy. It is recovered as the lawful dynamics of a coherent continuity branch. The coherent action gives a conserved density current and a quantum Hamilton-Jacobi equation. On the domain where the amplitude does not vanish, the phase one-form is locally exact. Its global periods are integer winding numbers, giving circulation in units of ℎ/𝑚. Those integers persist under every continuous admissible evolution and can change only when a defect crosses the return path or the coherent amplitude vanishes. The apparent discontinuity of a quantum phase slip is thus the topological consequence of continuous evolution through a configuration where the phase ceases to be defined. The same structure returns the exact Madelung representation of a selected Schrödinger branch. On configuration space it yields the Born current, current velocity and quantum potential. Adding an actual configuration following that velocity is the Bohmian ontological completion, not an unstated consequence of continuity. Equivariance then proves persistence of quantum equilibrium. At field level, a shift-symmetric action closes a classical current, its quadratic perturbations recover a characteristic metric, and stationary canonical data produce a recoverable bosonic quantum state. Gauge selection, spin, fermionic statistics, outcome selection and Standard Model promotion remain separately indexed descendants. The paper also closes the non-ideal boundary. Multiplying a density by an arbitrary positive weight generates an exact continuity source and an exact transformed quantum potential; it cannot be presented as a free admissibility correction. Any departure from the closed branch requires a declared self-adjoint operator or trace-preserving open-system law. Admissibility, Recoverability and Persistence then distinguish a genuine quantum branch from incomplete phase data, singular nodal flow or an unreturned measurement model. The result closes coherence, quantised circulation and quantum response within the recovered tree of nature as such. Keywords: coherence; phase; circulation; quantum potential; Madelung transform; Schrödinger equation; Bohmian guidance; equivariance; quantum field reconstruction; Chronoflux Atlas
We develop a mathematical framework in which physical description is organized through predictively sufficient closure of richer realization spaces. A closure map identifies realizations that are equivalent relative to specified observables, while dynamic sufficiency requires the induced reduced state to support autonomous evolution. This yields a precise formulation of the principle that law can close before realization does: a law-level state may be fully predictive within a declared domain even when multiple underlying realizations remain possible. We distinguish quotient closure, realization fibers, idempotent closure maps, refinement, partial closure, closure memory, obstruction, and saturation of multiple closure constraints. These structures are then tested against explicit physical examples involving compact phase winding, topological defects, gauge compensation, exchange statistics, spinorial closure, quantum measurement channels, and effective propagation geometry. The analysis exposes several limitations of a motivating one-scalar coherence construction. In particular, a smooth exact phase connection has vanishing local curvature; a relative phase does not generate orthogonality or Pauli exclusion; winding alone does not supply spin- or the observed electric-charge spectrum; and a single scalar gradient cannot define a generic nondegenerate four-dimensional metric. Rather than abandoning the coherence hypothesis, these obstructions motivate a minimally enriched effective architecture involving amplitude/phase, gauge connection, spinor, and geometric frame sectors. We distinguish this effective reconstruction from the stronger hypothesis that all such structures descend from a common parent coherence object. The resulting framework does not claim a completed derivation of the Standard Model, general relativity, or quantum measurement. Instead, it identifies the mathematical and physical gates such derivations must satisfy and isolates noncommuting saturation, closure obstruction, representation reconstruction, stable particle sectors, and universal emergent geometry as principal theorem and physics programs. Keywords: closure mathematics; predictive sufficiency; quotient dynamics; gauge theory; topological defects; particle identity; quantum foundations; emergent geometry; noncommuting closure; unification.