ground-state problem resides entirely in the $\pi^{od}$ of [[**\[pi-od\]**]{}]{}. We extract the sign $s(ii'{{\bf{k}}})$ of the coefficient $C_{ii'{{\bf{k}}}}$ in [[**\[pi-od\]**]{}]{} and, relating its magnitude to the joint $N$-OP $p_{11\cdots 1}(ii'{{\bf{k}}}) \equiv C_{ii'{{\bf{k}}}}^2$, we rewrite [[**\[pi-od\]**]{}]{} as $$\begin{aligned}
\pi^{od}(x'_1x'_2;x_1x_2) =& 
\sum_{\scriptscriptstyle
\overset{i<i'\ne j<j',{{\bf{k}}}}
{i,i',j,j' \not\in {{\bf{k}}}}}
s(ii'{{\bf{k}}}) s(jj'{{\bf{k}}}) \nonumber \\
&p^{1/2}_{11\cdots 1}(ii'{{\bf{k}}}) 
p^{1/2}_{11\cdots 1}(jj'{{\bf{k}}})\nonumber\\
&\left(
\psi_i(x'_1)\psi_{i'}(x'_2)-\psi_{i'}(x'_1)\psi_i(x'_2)\right)\nonumber\\
&\left(
\psi_j(x_1)\psi_{j'}(x_2)-\psi_{j'}(x_1)\psi_j(x_2)\right).
\label{pi-od-signs}\end{aligned}$$ We suppose that a variational approximation exists in which $$s(ii'{{\bf{k}}}) s(jj'{{\bf{k}}}) = s(ii') s(jj'), \forall{{\bf{k}}}.
\label{signconjecture}$$ The [*sign conjecture*]{} [[**\[signconjecture\]**]{}]{} reduces the sign complexity to algebraic, scaling as $M^2$. $\pi^{od}$ simplifies to $$\begin{aligned}
\pi^{od}(x'_1x'_2;x_1x_2) &=& 
\sum_{\scriptscriptstyle
{i<i' \ne j<j'}}
s(ii') s(jj') \nonumber \\
&&\left[ \sum_{\scriptscriptstyle \overset{{{\bf{k}}}}
{i,i',j,j' \not\in {{\bf{k}}}}}
p^{1/2}_{11\cdots 1}(ii'{{\bf{k}}}) 
p^{1/2}_{11\cdots 1}(jj'{{\bf{k}}}) \right] \nonumber\\
&&\left(
\psi_i(x'_1)\psi_{i'}(x'_2)-\psi_{i'}(x'_1)\psi_i(x'_2)\right)\nonumber\\
&&\left(
\psi_j(x_1)\psi_{j'}(x_2)-\psi_{j'}(x_1)\psi_j(x_2)\right).
\label{pi-od-signs1}\end{aligned}$$ The quantities $p^{1/2}_{11\cdots 1}(ii'{{\bf{k}}})$ and $p^{1/2}_{11\cdots
  1}(jj'{{\bf{k}}})$ are ${{\bf{k}}}$-th components of vectors ${{\bf{p}}}^{1/2}_{11\cdots 1}(ii')$ and ${{\bf{p}}}^{1/2}_{11\cdots 1}(jj')$. The bracketed quantity in [[**\[pi-od-signs1\]**]{}]{} is their scalar product. Express it as $$\begin{aligned}
\sum_{\scriptscriptstyle \overset{{{\bf{k}}}}
{i,i',j,j' \not\in {{\bf{k}}}}}
p^{1/2}_{11\cdots 1}(ii'{{\bf{k}}}) 
p^{1/2}_{11\cdots 1}(jj'{{\bf{k}}}) &=& \nonumber \\
p^{1/2}_{1100}(ii'jj') p^{1/2}_{0011}(ii'jj') \xi(ii'jj'),
\label{scalar}\end{aligned}$$ where $p_{1100}(ii'jj')$ is the square magnitude of the vector ${{\bf{p}}}^{1/2}_{11\cdots 1}(ii')$ and $p_{0011}(ii'jj')$ that of ${{\bf{p}}}^{1/2}_{11\cdots 1}(jj')$. $p_{1100}(ii'jj')$ is the probability that $\psi_i$ and $\psi_{i'}$ are occupied while $\psi_j$ and $\psi_{j'}$ are not: $$\begin{aligned}
p_{1100}(ii'jj') &=& 