Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities

What the math gives to ML

The paper turns exclusion of simultaneous zeros of two quadratic forms into a constructive positive-definite pencil certificate: for n >= 3, two homogeneous equations have no common nonzero solution exactly when some nontrivial linear combination of their matrices is positive definite. This gives neural networks a way to design pairs of quadratic feature constraints that cannot become jointly degenerate, rather than merely penalizing observed violations. The most direct transfer is a regularized quadratic feature block whose matrices are trained together with an explicitly optimized separating pencil and a spectral-margin loss. The guarantee is structural and applies to every hidden representation, while the engineering tradeoff is the cost of maintaining or estimating the smallest eigenvalue of the pencil.