Finite-Energy Stable Solitonic Solutions in the Madelung Quantum Fluid via Root-Free Spectral Inversion: Circumventing Derrick's Theorem and Topological Vortex Knot Particle Spectrum

Abstract:A long-standing obstacle in the mathematical physics of extended field configurations is Derrick's non-existence theorem (1964), which proves that static scalar solitons in spatial dimensions d ≥ 3 are intrinsically unstable against either infinite dispersive expansion or catastrophic singularity collapse. In this work, we resolve this foundational challenge by constructing a family of three-dimensional, finite-energy, nonlinearly stable solitary solutions within the hydrodynamical Madelung quantum continuum. Key Findings and Theoretical Contributions:1. Deductive Variational Formulation: From a unified action in canonical Clebsch variables, we rigorously derive the continuity equation, the generalized Bernoulli equation featuring the Bohm Quantum Potential Q = -(ħ²/2m) ∇²(√ρ)/√ρ, and cross-coupled convective gauge currents j^μ = q ρ v^μ.2. Evasion of Derrick's Theorem in 3D: We prove analytically through virial scaling that the quantum dispersive pressure of Q scales as λ², creating an absolute potential well in the virial energy functional that arrests collapse and establishes Lyapunov stability.3. Finite-Energy Proof & Conserved Charge: We numerically integrate the exact 3D stationary radial profile, establishing a strictly finite Hamiltonian energy (E = 15.7445 < ∞) and conserved fluid mass (N = 24.0364 < ∞).4. Nonlinear Dynamic Verification via Root-Free Resolvent: Using a continuous Root-Free Laplace spectral resolvent on Chebyshev nodes, we demonstrate dynamic stability under large (+15%) non-perturbative core amplitude shocks, showing damped Lyapunov oscillations without numerical stiffness blow-up.5. Topological Taxonomy of Elementary Particles as Vortex Knots (Hopfions): Building on the Hopf fibration π₃(S²) ≅ ℤ, we map the particle spectrum: the Higgs boson as the scalar mode (T_0,0), leptons (electron) as unknotted vortex rings (T_1,0) with half-integer spin J = ħ/2 arising from the topological Berry phase, and baryons (proton) as trefoil knots (T_2,3), geometrically proving Asymptotic Quark Confinement and explaining the proton-to-electron mass hierarchy (m_p / m_e ~ 10³) from minimal knot ropelength without parameter fine-tuning. Preprint prepared for submission to: Acta Physica Polonica B (Jagiellonian University / Polish Physical Society).