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.