Proposition: Which total order needs two rod legs

Through common total order $s^2$, that contraction is only

One noncompact derivative leg can enter at order $s^3$, and two can enter at order $s^4$. Coefficientwise products of the separate quadratic coefficients still require their own noncompact domains.

Proof. Multiply the derivative expansions. The linear-linear term has degree two, the linear-quadratic terms degree three, and the quadratic-quadratic term degree four. The linear functional $L_f$ has a compact smooth test kernel after integration by parts. Noncompact rod tails occur in the quadratic derivatives.