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
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
This note consolidates seven theorem-grade Wallstrom closure results for the SGOC programme's density-phase framework, and precisely identifies the two remaining open problems. The Wallstrom problem is the failure of the Madelung hydrodynamic reformulation of quantum mechanics to automatically enforce the integer quantisation of circulation of the quantum phase. The present note does not prove unrestricted Wallstrom closure. It maps the problem branch by branch. Seven branches are closed: (i) Primary single-valued complex spectral branches: Wallstrom quantisation is automatic on every connected nodal-complement component. (ii) Flat solid-torus collar branch: the condition reduces to one period on the core generator, with a gauge-covariant integer winding-number refinement. (iii) Contractible carrier windows: local face-curvature data determine local abelian edge-phase gauge classes. (iv) Closed surface carriers with global carrier data: the downstream abelian complement obstruction collapses. (v) Boundary torus of the flat toral collar branch: strong-global carrier transport is automatic and the branch data rigidify the carrier architecture up to gauge. (vi) Genus-zero carriers with local data and proper-support localisation: the ambient period torsor disappears and the discrete carrier branch closes canonically up to gauge. (vii) Radial discovery-direction spectral branch: the recovered nodal-free ground state closes immediately. The two remaining open problems are: the state-space selection theorem for the primary complex branch, and the phase-to-carrier transport/localisation theorem for the discrete carrier branch. Keywords: Wallstrom problem, Madelung hydrodynamics, quantum phase quantisation, carrier transport, holonomy, SGOC, circulation quantisation Related Programme: Spectral Geometry of Coherence (SGOC)
This record contains Version 2.0 of The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background. Version 2.0 supersedes the earlier narrower local formulation (DOI: 10.5281/zenodo.19118007). The earlier record worked within a restricted quadratic-gradient density class. The present record proves the same result on the full local first-order homogeneous density class, with the weighted bulk-plus-flux bridge as the load-bearing structural principle rather than a minimality assumption on derivative order. The theorem lane is therefore broader, the claim boundaries are cleaner, and the companion-note dependence has been updated to reflect the current SGOC corpus. This record DOI. 10.5281/zenodo.19189901 should be treated as the source-of-record citation for the general fixed-background Schrödinger result. The paper proves a narrow but exact claim: within the full first-order homogeneous density class compatible with continuity-current structure, the Fisher-corrected Madelung action is the unique admissible action family up to the stated constant, and its Euler–Lagrange equations are exactly the Schrödinger equation. The decisive density correction is forced inside an upstream capacity-geometric theorem base rather than inserted by a diffusion or stochastic postulate. The logical scaffold is explicit and limited. Part 1A fixes the canonical logarithmic capacity scalar and the unique quadratic base tension. Parts 6 and 7A isolate the phase-blindness ceiling for the continuous quadratic sector. Part 6A sharpens the localisation-versus-smoothing and forced-balance framework. The present paper then shows that continuity-current compatibility fixes the phase sector, locality and first-order dependence force the full homogeneous density family, and a universal weighted bulk-plus-flux bridge with field-independent coefficients forces the Fisher form. The canonical member of that family is fixed by the bridge to the imported capacity scalar. In that precise sense the logical order at the Schrödinger layer is reversed: the geometric base comes first, and the Schrödinger action appears downstream of it. The claim is exact within its lane and no broader. This is a fixed-background, single-particle theorem stated away from nodal sets. It does not claim a many-body theory, a measurement theory, a relativistic completion, a full nodal-complement reconstruction theorem, or backreaction of the quantum sector on the background geometry. Appendix D records that on a nodal-complement component, once the phase-sensitive completion is an S¹-valued phase field, circulation quantisation and wavefunction reconstruction are automatic. Appendix G records a conditional H¹-closure lane inside the same fixed-background action. The discrete-curvature and branch-bookkeeping support used on nodal-complement triangulations belongs to the companion Closure Phase note, while the further complement-globalisation step belongs to the companion Relative Cohomology note. Keywords: Schrödinger equation; capacity geometry; Fisher information; Madelung action; phase-sensitive evolution; forced action; homogeneous density class; weighted Fisher bridge; REG; coherence scalar; fixed background; nodal complement; reproducibility Companion manuscripts cited in this paper's proof chain, from the SGOC programme: Part 1A: The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge. DOI: 10.5281/zenodo.19156316 Part 7A: The Second-Order Ceiling, Triadic Completion, and the Wilson-Triangle Lift. DOI: 10.5281/zenodo.19175902 Closure Phase as Discrete Curvature on a Triangulated 2-Complex. DOI: 10.5281/zenodo.19185885 Relative Cohomology and Cut-Graph Holonomy Trivialisation. DOI: 10.5281/zenodo.19185877 Additional provenance notes cited in the paper Part 6A: Localisation Rather than Saturation in Capacity Geometry: REG Dissipation, Forced Balance Laws, and a Restricted Ricci Bridge. DOI: 10.5281/zenodo.19174434 Landauer, Relative Entropy, and Fisher Length in Capacity Geometry. DOI: 10.5281/zenodo.19186309
This record contains The Ledger Hamiltonian on a Capacity Background, a standalone companion note within the Spectral Geometry of Coherence (SGOC) programme. The paper studies a distinct theorem lane from the general Schrödinger paper. Its focus is the chart-covariant first-order Hamiltonian and continuity structure induced by a fixed capacity background, rather than the density-phase forced-action classification treated in The Schrödinger Equation from Capacity Geometry. Working on the phase chart, the note identifies the natural first-order self-adjoint Hamiltonian representative associated with exact advection, tracks its measured-chart form, and records the corresponding continuity law, current, and falsifier structure. The claim is exact within its lane and no broader. This is a fixed-background, first-order transport result. It does not by itself claim the forced density-phase action theorem of the separate Schrödinger paper, and it does not claim a many-body theory, measurement theory, relativistic completion, or backreaction of the transported sector on the background geometry. Within the SGOC programme, this note should be read as complementary to, but distinct from, The Schrödinger Equation from Capacity Geometry. The present paper isolates the chart-covariant Hamiltonian generator and continuity structure on the phase chart; the Schrödinger paper addresses the separate fixed-background density-phase action lane. The note also serves as a conceptual bridge to the downstream hydrogen toy-model note, where the fixed-background framework is used as a spectral probe. Keywordscapacity geometry; Hamiltonian; exact advection; continuity law; phase chart; fixed background; transport generator; SGOC Related identifiers Cites -> 10.5281/zenodo.19156316The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge IsSupplementTo -> 10.5281/zenodo.19189901The Schrödinger Equation from Capacity Geometry IsSupplementTo -> 10.5281/zenodo.19191108The Hydrogen Atom on a Capacity Background
This record contains The Hydrogen Atom on a Capacity Background, a fixed-background hydrogenic toy-model note within the Spectral Geometry of Coherence (SGOC) programme. The paper is not a general derivation of quantum mechanics. Instead, it uses the imported capacity scalar and fixed-background framework as a controlled hydrogenic spectral and inverse-probe test bed. Its aim is to show how the capacity-background formalism can be used to organise, interpret, and recover hydrogenic structure in a mathematically disciplined toy-model setting. In that sense, it is a downstream application note rather than a foundational theorem paper. The claim is exact within its lane and no broader. This is a fixed-background, single-particle, hydrogenic toy-model analysis. It does not claim a relativistic completion, a many-body theory, a full measurement theory, or backreaction of the quantum sector on the background geometry. Nor does it replace the separate fixed-background action theorem proved in The Schrödinger Equation from Capacity Geometry. Within the SGOC programme, the paper should be read as a downstream companion to the foundational capacity-geometry notes and the general Schrödinger paper. Part 1 and Part 1A supply the intrinsic background conventions and the canonical capacity scalar, Part 7A supplies the phase-blindness ceiling relevant to the scope discussion, and the general Schrödinger paper supplies the separate phase-sensitive fixed-background action lane. The present note then uses that broader framework as a hydrogenic test case. Keywordshydrogen atom; capacity geometry; fixed background; spectral probe; inverse problem; single-particle model; SGOC Related identifiers Cites -> 10.5281/zenodo.19156067Capacity Geometry: REG, Dirichlet Tension, and Quasi-Local Mass Cites -> 10.5281/zenodo.19156316The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge Cites -> 10.5281/zenodo.19175902The Second-Order Ceiling, Triadic Completion, and the Wilson-Triangle Lift IsSupplementTo -> 10.5281/zenodo.19189901The Schrödinger Equation from Capacity Geometry Optional: IsSupplementTo -> 10.5281/zenodo.19189960The Ledger Hamiltonian on a Capacity Background
The Schrödinger Equation from Capacity Geometry The Schrödinger equation is usually either taken as a starting postulate or recovered by adding Fisher information or an exact-uncertainty principle as an independent quantum input. This paper proves a narrower but different result: within a declared fixed-background local action class, the nonrelativistic Schrödinger equation is forced downstream of an independently established geometric scalar rather than by postulating the Fisher term at the quantum layer. More precisely, within the declared local phase-independent quadratic-gradient density class, the Fisher-corrected Madelung action is the unique action family compatible with the admissible-class constraints, and its Euler–Lagrange equations are exactly the Schrödinger equation. The scaffold comes from two companion theorem lanes in The Spectral Geometry of Coherence programme, where Part 1A establishes the canonical logarithmic capacity scalar and the uniqueness of the quadratic base tension in the declared geometric class. And Part 7A proves that the continuous translation-invariant quadratic sector is phase blind, so a phase-sensitive evolution law cannot come from that sector alone. The present paper then shows that continuity-current compatibility fixes the phase sector, the quadratic-gradient and scaling assumptions force the Fisher-form density correction up to a constant, and the bridge to the imported scalar fixes the canonical member of that family. In that precise sense, the logical order is different at the Schrödinger layer: the geometric base comes first, and the Schrödinger action appears downstream of it. The claim is exact within its lane and no broader. It is a fixed-background, single-particle, local theorem, stated away from nodal sets. It does not claim a many-body theory, measurement theory, relativistic completion, global nodal reconstruction, or backreaction of the quantum sector on the background geometry. For programme context, readers can trace the scaffold through The Spectral Geometry of Coherence (SGOC), Volume Two, Parts 1–7, 10.5281/zenodo.18462286, together with Part 1A https://doi.org/10.5281/zenodo.19113748, Part 6A, 10.5281/zenodo.19113871, and Part 7A, 10.5281/zenodo.19113876.
The Spectral Geometry of Coherence (SGOC) is a research programme asking whether one forced geometric scalar can act as a common interface between quantum mechanics, gravity, and arithmetic structure. The central object is a scalar field called Sc, Sc = −log(1 − R²), where R measures the normalized coherence between two coupled channels. Sc is not chosen arbitrarily. It is the coordinate forced by the programme’s own transport law: multiplicative slack becomes additive capacity. From this scalar, together with the declared fixed-background action class, SGOC derives the nonrelativistic Schrödinger equation as a forced geometric consequence rather than taking it as a starting postulate. The same scalar also controls capacity-metric curvature and satisfies an exact identity linking three objects normally discussed in separate settings: the quantum potential, scalar curvature, and an Unruh-type temperature. This is the Jacobson-Schrodinger compatibility law. The most recent arithmetic results show that, on a selected mathematical branch, the same scalar is also a zeta scalar: Sc = log ζ(σ). In that branch, Sc is a lossless register of prime-power source data. A conditional theorem then connects prime source data, through canonical Sc, to a selected tensorial Plebanski-Einstein gravity branch. The theorem is not a claim that primes vaguely “look like” gravity; it is a conditional chain with stated hypotheses and failure modes. The Wallstrom problem, a long-standing obstacle for geometric derivations of quantum mechanics, is addressed directly. Multiple branches are resolved under stated hypotheses, while the remaining open problems are named rather than hidden. The programme is large and modular, explicit about boundaries, and covers many papers and more results of substance than the selected highlights above shows. Some results are clean theorems. Others are conditional on named assumptions. Some are numerical audits with stated controls. Several attractive routes are ruled out by negative results, including a density-collapse result showing that smoothed scalar density alone cannot carry the full arithmetic content. All results are provisional pending independent expert review. This document is a pedagogical companion to the fully technical version: SGOC Programme Compendium – Technical, 10.5281/zenodo.19614677