| _version_ | 1866902103296835584 |
|---|---|
| author | Cobb, Stephen Euin |
| author_facet | Cobb, Stephen Euin |
| contents | <p>We derive a unified field equation for the four-dimensional curvature continuum introduced in earlier papers of this series (R42–R45). Within this medium, all particles and forces arise as standing or traveling modes of rotation and torsion on the hypersphere S³. The derived <strong>rotor–wave equation</strong></p> <p> ρ ∂²_t Ψ − κ ∇×(∇×Ψ) + ∂U/∂Ψ = 0</p> <p>governs both curvature amplitude and plane orientation, coupling mass and charge naturally through dual-plane geometry. Linear and sectoral limits of this equation reproduce the Maxwell, Dirac, Proca/Yang–Mills, and Schrödinger forms, showing that standard quantum and gauge theories are low-amplitude projections of a single nonlinear field law. Eigenmode closure on S³ quantizes mass, spin, and charge through the topological parameters (N, n, θ_eff, Δφ₄). The resulting scaling relations—<em>m ∝ N³ / R²</em>, <em>g = 2 sec θ_eff</em>, <em>g_A = ± sec Δφ₄</em>—match the observed electron, proton, neutron, and baryon families to within experimental precision. The constants <em>c</em>, <em>ħ</em>, and <em>α</em> emerge as ratios of the medium’s stiffness κ and torsional density ρ, while conservation of energy, momentum, charge, and baryon number follows from curvature continuity. Numerical eigenmode solutions on S³ are proposed to generate predictive mass ladders and magnetic–axial correlations, offering falsifiable tests across leptons, baryons, mesons, and bosons. The theory unites all known particles and interactions as harmonic excitations of one self-consistent geometric field—a continuum whose internal motion constitutes both matter and radiation.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17334381 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Final Equation: Unifying Maxwell, Dirac, Proca/Yang–Mills, and Schrödinger Cobb, Stephen Euin four-dimensional curvature field rotor–wave equation S³ topology unified particle spectrum lepton–baryon unification magnetic and axial couplings geometric quantization curvature stiffness torsional waves field–geometry correspondence continuous medium theory <p>We derive a unified field equation for the four-dimensional curvature continuum introduced in earlier papers of this series (R42–R45). Within this medium, all particles and forces arise as standing or traveling modes of rotation and torsion on the hypersphere S³. The derived <strong>rotor–wave equation</strong></p> <p> ρ ∂²_t Ψ − κ ∇×(∇×Ψ) + ∂U/∂Ψ = 0</p> <p>governs both curvature amplitude and plane orientation, coupling mass and charge naturally through dual-plane geometry. Linear and sectoral limits of this equation reproduce the Maxwell, Dirac, Proca/Yang–Mills, and Schrödinger forms, showing that standard quantum and gauge theories are low-amplitude projections of a single nonlinear field law. Eigenmode closure on S³ quantizes mass, spin, and charge through the topological parameters (N, n, θ_eff, Δφ₄). The resulting scaling relations—<em>m ∝ N³ / R²</em>, <em>g = 2 sec θ_eff</em>, <em>g_A = ± sec Δφ₄</em>—match the observed electron, proton, neutron, and baryon families to within experimental precision. The constants <em>c</em>, <em>ħ</em>, and <em>α</em> emerge as ratios of the medium’s stiffness κ and torsional density ρ, while conservation of energy, momentum, charge, and baryon number follows from curvature continuity. Numerical eigenmode solutions on S³ are proposed to generate predictive mass ladders and magnetic–axial correlations, offering falsifiable tests across leptons, baryons, mesons, and bosons. The theory unites all known particles and interactions as harmonic excitations of one self-consistent geometric field—a continuum whose internal motion constitutes both matter and radiation.</p> |
| title | The Final Equation: Unifying Maxwell, Dirac, Proca/Yang–Mills, and Schrödinger |
| topic | four-dimensional curvature field rotor–wave equation S³ topology unified particle spectrum lepton–baryon unification magnetic and axial couplings geometric quantization curvature stiffness torsional waves field–geometry correspondence continuous medium theory |
| url | https://doi.org/10.5281/zenodo.17334381 |