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 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.
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
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 isolates the free fixed-background sector of the capacity-geometric Schrodinger action and proves that it already carries the full standard nonrelativistic free-particle package. In density-phase variables the action is Galilei covariant with the exact boost law for rho and S; the constant-density plane-wave sector reproduces the standard dispersion E = |p|^2/(2m); and the free evolution operator has the usual oscillatory kernel together with the correct composition law. The stationary-phase analysis of that kernel recovers the classical straight-line ray with momentum p* = m(x-x0)/t. The novelty is not free-particle quantum mechanics itself. The genuine contribution is a hard consistency check: once the SGOC geometric inputs are fixed, the free sector lands on the standard boost law, standard dispersion, and standard propagator without further tuning, extra correction terms, or a hidden preferred frame. For a derivation programme, that is a real rigidity test rather than a decorative benchmark. The claim boundary is narrow and explicit. This is a fixed-background, force-free note. It does not derive new physics, does not add interactions, and does not try to reinterpret the free kernel beyond showing that the SGOC action reproduces the ordinary nonrelativistic structure exactly where it should. Within SGOC, this is the baseline sanity-check paper for the quantum lane. It confirms that the fixed-background forced-action theorem is not only formally equivalent to Schrodinger evolution on paper, but also reproduces the operational free-sector structures that any credible downstream extension must preserve. For mathematical physicists, this matters because derivation programmes often fail at the simplest covariance or kernel tests. Here the basic free package survives intact. That makes the later electromagnetic, spin, and relativistic extensions scientifically worth reading rather than suspect from the outset. Keywords: Schrodinger equation, Galilean covariance, free propagator, plane waves, capacity geometry, SGOC, free sector. DOI this paper: 10.5281/zenodo.19593108. Related SGOC papers: - The Schrodinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background - Local U(1) Gauge Covariance, Minimal Electromagnetic Coupling, and Gauge-Covariant Spectral Recovery on a Capacity Background - A Pauli-Type Spinor Completion of the Capacity-Geometric Schrodinger Action Programme summary files - before and beyond Volume Three SGOC Programme Compendium - Pedagogical 10.5281/zenodo.19614652 SGOC Programme Compendium - Technical 10.5281/zenodo.19614677
This note studies the first fixed-background spinor completion of the minimally coupled SGOC kinetic sector. If one requires a rank-two, rotation-covariant, locally U(1)-gauge-covariant square-root completion generated by sigma dot Pi, then the Pauli identity forces the spin-magnetic coupling exactly. The resulting Hamiltonian is the standard Pauli Hamiltonian on a capacity background with scalar potential q Phi + V_phys - alpha Sc + eta, and the corresponding action yields the Pauli equation by direct variation. The novelty is not the Pauli equation itself, nor the existence of spin. The paper's real contribution is a rigidity statement inside the SGOC class: after the scalar and electromagnetic sectors are fixed, the first admissible rank-two spinor completion has no freedom left in the Zeeman coefficient. The standard spin-magnetic term is forced rather than tuned. This is still a fixed-background extension note. It does not derive spin from a deeper ontology, does not address anomalous magnetic moments beyond the tree-level g = 2 value, and does not attempt a full relativistic field theory. Its scope is the first admissible spinor completion of the SGOC Schrodinger action. Inside SGOC, this note closes the nonrelativistic spin lane and sets up the relativistic continuation. It is the natural successor to the minimally coupled U(1) paper and the immediate predecessor of the Dirac completion, so it helps show that the programme can upgrade stepwise without losing internal constraint. For the wider science community, the paper matters because spin is not a cosmetic add-on. Any framework that hopes to speak seriously to atomic or condensed-matter physics has to land on the Pauli sector cleanly. Here it does so through an algebraic forcing argument rather than by parameter choice. Keywords: Pauli equation, spinor completion, g-factor, Zeeman term, Pauli-Landau spectrum, U(1) gauge covariance, capacity geometry, SGOC. DOI this paper: 10.5281/zenodo.19593088. Related SGOC papers: - Local U(1) Gauge Covariance, Minimal Electromagnetic Coupling, and Gauge-Covariant Spectral Recovery on a Capacity Background - A Dirac-Type Relativistic Completion of the Capacity-Geometric Schrodinger Action - Overdetermined Gauge-Covariant Spectral Recovery and Held-Out Prediction on Electromagnetic and Spin-Polarised Pauli Backgrounds Programme summary files - before and beyond Volume Three SGOC Programme Compendium - Pedagogical 10.5281/zenodo.19614652 SGOC Programme Compendium - Technical 10.5281/zenodo.19614677