NP-hardness of globally optimal nonnegative rank-two approximation

The paper gives a polynomial-time many-one reduction from positive one-in-three
satisfiability to the exact rational-input decision problem for nonnegative
rank-two approximation in the squared Frobenius norm, so that this problem is
NP-hard. The reduction produces strictly positive, symmetric positive definite
rational matrices whose unconstrained best rank-two approximation error is an
explicit rational threshold; a nonnegative approximation attains that
threshold exactly when the Boolean instance is satisfiable. The proof uses a
rational orthogonal projector to encode the Boolean equations in a repeated
eigenspace and a quantitative endpoint-rounding argument to recover a Boolean
solution. The reduction has an inverse-polynomial additive error gap, and a
rational perturbation makes the spectrum simple while preserving hardness.
Membership of the problem in NP is not asserted.