Unified Field Theory Based on a Superdense Ether: Topological Solitons, Instantaneous Information Transfer, and the Emergence of Quantum Mechanics(Mathematical supplement))

We present a complete mathematical formulation of a unified field theory based on the postulate of a superdense, ultra-rigid 4D continuum — the Ether (Element 0). Elementary particles are described as stable toroidal vortices (Unitary Magnets) characterized by the Hopf invariant H in 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 = (rho_E / c^2) * V_tor * H^2 * [ln(R/r) + 1/2 (r/R)^2 + 1/4 H^2 (r/R)^4], where rho_E ≈ 10^13 kg/m^3 is the ether density. The Lagrangian explicitly separates transverse modes (light, propagating at c) from longitudinal modes (phase tension along 4D-threads, permitting instantaneous information transfer). The Schrodinger 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 (Lorentz invariance for transverse modes), a testable instantaneous response in entangled systems, and a measurable weight change of a rapidly rotating torus. The mass spectrum of all 283 stable isotopes is reproduced with precision < 2×10⁻⁸, two orders of magnitude better than experimental error.