\label{E-od}\end{aligned}$$ where ${\cal K}_{ii',jj'} = (\psi_i\psi_{i'},\hat{w}\psi_j\psi_{j'})$. [[**\[Esum\]**]{}]{} – [[**\[E-od\]**]{}]{} define the OP-NOFT energy functional. Including the complexity of efficient evaluation of the matrix elements, it scales as $M^5$ if the $N$-representability conditions for the 3- and 4-OPs can be limited to those deriving from the $(3,q)$ and $(4,q)$ positivity conditions with $q
\le 4$.

Proof of the sign conjecture
----------------------------

A variational sign approximation must be a statement about the sign $s(ii'{{\bf{k}}})$ or $s(jj'{{\bf{k}}})$ of each coefficient appearing in [[**\[pi-od\]**]{}]{}. To prove the sign conjecture [[**\[signconjecture\]**]{}]{}, we must find one in which the ${{\bf{k}}}$-dependences of $s(ii'{{\bf{k}}})$ and $s(jj'{{\bf{k}}})$ cancel out. One, for the general case of matrix elements $\left[ {\cal K}_{ii',jj'} - {\cal K}_{ii',j'j} \right]$ of arbitrary sign, is presented here. Another, valid only for positive matrix elements, is presented in Section S1 of the Supporting Information (SI). The two approximations yield the same results for OP-NOFT-0.

[*Arbitrary signs of matrix elements:*]{} Assigning each index $l$ in $C_{{{\bf{k}}}}$ a sign $s(l)$ and taking $s({{\bf{k}}})$ as their product, $$s({{\bf{k}}}) = \prod_{l \in {{\bf{k}}}} s(l),$$ is a variational approximation. Consequently $s(ii'{{\bf{k}}}) = s(i) s(i')
s({{\bf{k}}})$ and $$s(ii'{{\bf{k}}}) s(jj'{{\bf{k}}}) = s(i) s(i') s(j) s(j'),
\label{signprod}$$ and the sign conjecture is proved for arbitrary 