Emergence of Quantum Mechanics and Topological Soliton Stability in a Saturated Cubic Quintic Cosserat Korteweg Continuum

The operational success of quantum mechanics is undisputed, yet the search for a deterministic, mechanical ontology underlying its probabilistic formalism remains an open challenge. Building upon the framework of hydrodynamic quantum analogs, this paper proposes a classical deterministic model wherein wave-particle duality and quantization emerge exactly from the continuum mechanics of a generalized micropolar fluid. We define a Cosserat-Korteweg continuum governed by a geometric virial expansion that inherently produces a Cubic-Quintic rheology: a cubic attractive term representing Bernoulli dynamic suction, and a quintic repulsive term representing a strict hard-core density saturation limit (a jamming phase). By incorporating Fisher Information as a thermodynamic capillarity penalty within the fluid’s Master Lagrangian, we analytically derive the Bohmian quantum potential as an exact elastodynamic stress. Applying the Madelung transformation to the irrotational asymptotic limit of this substrate, we demonstrate that the non-linear macroscopic evolution of the medium is strictly governed by the Cubic-Quintic Nonlinear Schrödinger Equation (CQ-NLSE). We prove that the quintic repulsion term prevents the Townes singularity collapse, satisfying the Vakhitov-Kolokolov criterion and ensuring the absolute thermodynamic stability of topological solitons. Finally, we demonstrate that the Heisenberg Uncertainty Principle is not an axiomatic limit on determinism, but a rigorous geometric consequence of the Cramér-Rao bound applied to the continuum’s topological hydrostatic limits.