The Gemini-Ribé Complex Tensor Formulation: Topological Rescue of Conservation Equations under Extreme Gradient

Continuous modeling of fluids, elastic solids, magnetohydrodynamic plasmas, and elec-tromagnetic fields under extreme spatiotemporal gradients historically encounters nonlinearclosure instability. In high-turbulence hydrodynamics or wave shoaling, the expansion ofthe fourth-order moment tensor to 81 dimensions (R81 ) and its subsequent collapse to Mil-lionshchikov’s 15-dimensional quasi-normal manifold (R15 ) leads to the so-called Ogura’sCurse, characterized by the appearance of negative spectral energy densities and instanta-neous numerical collapse (NaN). This work presents the unified theoretical framework of theRibé Topological Rescue. It is demonstrated that the state space of a 3D continuum is notrestricted to the classical 9-dimensional tensor, but incorporates 2 complex orthogonal dimen-sions for each physical spatial axis (2 × 3 = 6D), constituting an extended 15-dimensionalmanifold (C15 ). By considering the quadratic autocorrelation of the flow, the interactivespace expands to the tensor product of 15 × 15 = 225 dimensions (C225 ). The Ribé Reduc-tion and Saturation Theorem is formulated, analytically demonstrating that upon reachingthe symmetric stress limit capacity Ξcrit in the 120-dimensional symmetric subspace, therational saturation function Γ transfers 100% of the non-dissipable excess energy towardthe core of 105 orthogonal antisymmetric dimensions (iΞantisym ). This mechanism actsas a 90◦ rotation operator, collapsing the 225 interactive dimensions back to the stable15-dimensional manifold (C225 → C15 ) and guaranteeing a strict spectral positivity of theenergy (E(k, t) ≥ 0∀k, t). The four extended fundamental laws are rigorously derived intheir Cartesian components and validated in HPC infrastructures against ballistic impacts,hyperbolic shocks, and astrophysical plasma reconnection.