Singular geometry and eigenframe topology in local rank-2 tensor observables

Symmetric second-rank tensors are reported through magnitude-ordered principal values and axes. This representation folds tensor space: although the physical tensor remains smooth, the reported parameters develop cusps and exchange labels when one principal value crosses zero or two become degenerate. It conceals a global effect: an arrow chosen along a principal axis and transported continuously around a closed loop encircling a degeneracy can return with opposite orientation, even though the tensor returns to itself. This reversal defines a binary return parity that is invariant under smooth loop deformations that avoid degeneracy. We use the electric-field-gradient (EFG) tensor as a clean prototype because it is symmetric, traceless, and locally measurable at a probe site. Applied strain provides external control coordinates that steer the local EFG through tensor space, as determined from first-principles calculations. In rutile TiO2, we identify an isolated control-space degeneracy with nontrivial parity; SnO2 exhibits point- or line-like degeneracies depending on the control slice; and in cubic MgO, strain locally controls all five EFG components. These results distinguish local reporting singularities from global eigenframe topology. The underlying spectral geometry also applies to spatial tensor fields, and strain-tuned, orientation-resolved hyperfine spectroscopy offers a route to reconstruct return parity locally.