Tensor Operator Singularity is ∃R-Complete

The determinant gives a complete algebraic characterization of singularity for matrices. In multilinear algebra, there are several notions of degeneracy and rank, but no single determinant-like invariant applies uniformly across all tensor formats. We study Tensor Operator Singularity, a real multilinear feasibility problem associated with a third-order tensor. Given a tensor T ∈ Q r×n×p , it defines a parameterized family of linear maps L T (y) : R n → R r by contracting the third mode against y. The decision problem asks whether there exist nonzero x ∈ R n and y ∈ R p such that L T (y)x = 0. We prove that Tensor Operator Singularity is complete for the existential theory of the reals. The proof proceeds through an explicit chain of polynomialtime reductions from homogeneous quadratic feasibility to projective bilinear feasibility and then to Tensor Operator Singularity. The final reduction is coefficient-preserving: a bilinear system x ⊤ M ℓ y = 0 is encoded by the tensor T (ℓ, i, j) = (M ℓ) ij. We also give exact linear-matrix-space interpretations of the problem, clarify its relation to matrix singularity, tensor rank, tensor eigenvalue problems, hyperdeterminants, symbolic matrix singularity, and noncommutative rank, and record limitations and open directions.