The Equation of Everything (SEoT): Elastic Quantum Framework on Spherical DEM Compactifications

We present the spherical extension of the Equation of Everything (SEoT), a unified 4D elastic framework that integrates gauge, fermionic, and gravitational fields within the Elastic DEM–Spheres formalism.In this version, spacetime is compactified as R×S3\mathbb{R} \times S^3R×S3, turning the flat DEM background into a curved elastic manifold.This curvature introduces natural confinement, spectral gaps, and holographic properties consistent with Maldacena’s correspondence. 1. Overview In the SEoT, spacetime behaves as an elastic continuum endowed with stress–energy deformations that generate the fundamental interactions.The DEM–Spheres version extends this to compact 3-spherical geometries, where elastic curvature couples directly to quantum and cosmological observables.The model unifies: Tμν=λ gμν (∇αξα)+2μ ∇(μξν)+Qμν(R,Φ,ψ)T_{\mu\nu} = \lambda\, g_{\mu\nu}\, (\nabla_\alpha \xi^\alpha) + 2\mu\, \nabla_{(\mu}\xi_{\nu)} + \mathcal{Q}_{\mu\nu}(\mathcal{R}, \Phi, \psi)Tμν=λgμν(∇αξα)+2μ∇(μξν)+Qμν(R,Φ,ψ) Here, ξμ\xi_\muξμ represents the elastic displacement field, while Qμν\mathcal{Q}_{\mu\nu}Qμν encodes gauge and fermionic couplings within curved topology.The curvature of S3S^3S3 ensures finite vacuum rigidity, producing a mass gap consistent with the constructive proofs in Rigorous Constructive Framework for a Mass Gap in Elastic Quantum Models (Part I & II). 2. Holographic and Maldacena Interpretation In the spherical DEM setting, the SEoT acquires holographic correspondence:the elastic degrees of freedom in the 4D bulk map onto conformal boundary modes on S3S^3S3.This provides a dual elastic–conformal dictionary: Elastic stress in bulk↔Boundary curvature response (CFT)\text{Elastic stress in bulk} \quad \leftrightarrow \quad \text{Boundary curvature response (CFT)}Elastic stress in bulk↔Boundary curvature response (CFT) This bridge connects the mass-gap elasticity program with Maldacena’s AdS/CFT paradigm, offering a geometric realization of quantum gravity through elastic curvature. 3. Cosmological Implications The curved elastic framework naturally explains: The cosmological constant as a residual elastic tension in S3S^3S3. The accelerated cosmic expansion as relaxation of curvature stress. The unification of dark energy, dark matter, and vacuum elasticity under a single geometric law. The DEM–Spheres version also predicts small oscillations in curvature energy, potentially observable as Hubble tension modulations. 4. Relation to Previous Works This version supersedes the flat-space SEoT (v1.0), embedding its equations into the DEM–Spheres topology and connecting directly to: Part I: Flat DEM/QEM Mass Gap Framework. Part II: Spherical Compactifications and Curved Elastic Dynamics. Together, these establish a constructive and geometrically consistent Theory of Everything, bridging elasticity, quantum field theory, and cosmology. “This version also introduces a natural holographic realization of the SEoT through the curvature of S³×S³, unifying bulk elasticity and boundary dynamics.”