The Gemini-Ribé Complex Tensorial Formulation: Topological Rescue of Conservation Equations under Extreme Gradients

The continuum modeling of fluids, elastic solids, magnetohydrodynamic plasmas, andelectromagnetic fields under extreme spatiotemporal gradients historically encounters non-linear closure instability. In high-turbulence hydrodynamics or wave shoaling, the expansionof the fourth-order moment tensor to 81 dimensions (R81 ) and its subsequent collapse to theMillionshchikov quasi-normal 15-dimensional manifold (R15 ) leads to the so-called Ogura’sCurse, characterized by the emergence of negative spectral energy densities and instanta-neous numerical collapse (NaN). This work presents the unified theoretical framework ofthe Ribé Topological Rescue. It is demonstrated that the state space of a 3D continuum isnot restricted to the classical 9-dimensional tensor, but incorporates 2 complex orthogonaldimensions per physical spatial axis (2 × 3 = 6D), constituting an extended 15-dimensionalmanifold (C15 ). By considering quadratic flow autocorrelation, the interactive space expandsto the tensor product of 15 × 15 = 225 dimensions (C225 ). The Ribé Saturation-ReductionTheorem is formulated, analytically proving that upon reaching the symmetric stress limitcapacity Ξcrit , the rational saturation function Γ transfers 100% of the non-dissipable ex-cess energy towards the orthogonal antisymmetric kernel iΞantisym . This mechanism actsas a 90◦ rotation operator, collapsing the 225 interactive dimensions back to the stable15-dimensional manifold (C225 → C15 ) and guaranteeing strict spectral energy positivity(E(k, t) ≥ 0 ∀k, t). The four extended fundamental laws are rigorously derived: Navier-Stokes-Gemini-Ribé, Cauchy-Gemini-Ribé, MHD-Gemini-Ribé, and Maxwell-Gemini-Ribé,providing a fully reproducible reference method in Python.