Unified Field Theory Based on a Superdense Ether: Topological Solitons, Mass Spectrum, Quantum Emergence, and Unified Density Scale β = 10⁻⁴

We present a complete mathematical formulation of a unified field theory based on the postulate of a superdense, ultra-rigid 4D continuum — the Ether. Elementary particles are described as stable toroidal vortices (Unitary Magnets) characterized by the Hopf invariant H ∈ Z, a topological integer quantifying the linking number of phase threads. The mass spectrum is derived from the equations of motion of the elastic medium, yielding m = (ρE/c2)·Vtor ·H2 ·f(R/r) with f(R/r)=ln(R/r)+ 12(r/R)2 + 14H2(r/R)4. Drawing from V.P. Oleinik’s description of the electron as an open self-organizing system, we define the fundamental particle as a unitary magnet. We derive the fundamental Planck density ρP ≈ 5.15 × 1096 kg/m3 from first principles. ToresolvetheapparentdiscrepancybetweenρP ≈1096,observednuclearmatterdensity ρN ≈ 1017 kg/m3, and the effective density ρE ≈ 1013 kg/m3 used in the mass formula, we introduce a universal scaling constant β = 10−4. All densities are unified by ρ(k) = ρP · βk, where k is the topological reduction exponent. The Lagrangian explicitly separates transverse modes (light, propagating at c) from longitudinal modes (phase tension along 4D-threads, permitting instantaneous information transfer). The Schr ̈odinger equation with a nonlocal term is derived from the classical field theory, explaining quantum entanglement without violating special relativity. Experimental predictions include the absence of ether wind, a testable instantaneous response in entangled systems, and a measurable weight change of a rapidly rotating torus. The mass formula reproduces all 283 stable isotopes with relative error < 2 × 10−8. Keywords: ether, Hopf invariant, topological soliton, unitary magnet, quantum en- tanglement, unified field theory, instantaneous information transfer, Planck density, scaling relation, β = 10−4