Exact Lagrangian Realization and Robust Strain Sensing in Incompressible Flow

We prove that the full group \(\SL(3,\R)\) occurs as the set of one-particle deformation gradients of periodic, unforced, single-shell solutions of both the three-dimensional Euler and Navier--Stokes equations. More precisely, given a particle label \(p\in\T^3\), a time \(T>0\), and \(F_*\in\SL(3,\R)\), every sufficiently large odd integer \(N\) admits a real-analytic curl eigenfield \(W_N\) with \(\operatorname{curl}W_N=NW_N\). With an explicit scalar amplitude, \(W_N\) yields a steady Euler solution; with an explicit exponentially decaying amplitude, it yields a Navier--Stokes solution for any positive viscosity, and in both cases \(\nabla_aX(p,T)=F_*\). The lifted particle trajectory is an embedded analytic arc with nowhere-vanishing velocity. The construction combines global trace-free symmetric-matrix control on \(\SL(3,\R)\), a Beltrami Cauchy problem along the controlled arc, Runge approximation, inverse localization on the torus, and a finite-dimensional endpoint correction.
We also classify finite material-direction systems that determine every trace-free strain after arbitrary volume-preserving deformation. In dimension \(n\), this congruence-robust property holds exactly when the associated rank-one projectors span \(\Sym(n)\); hence the sharp number of scalar channels is \(n(n+1)/2\). In dimension three, among minimal systems the undeformed outer-product lower bound is maximized exactly by the six axes of a regular icosahedron. The realization theorem shows that the full deformation-group quantifier in this sensing result is dynamically attained within the rigid class above.