Theorem: Four exact pattern calculations

The matrix $K$ is positive semidefinite and

For $\delta {\mu }_s={\sum }_i{s}_i{f}_i$, $E_G(s)=s^{\mathsf{T}}Ks/2$. In particular:

1. aligned:

   $$E_{\mathrm{al}}=\frac{1}{2}\sum _{i,j}{K}_{ij};$$

2. alternating:

   $$E_{\mathrm{alt}}=\frac{1}{2}\sum _{i,j}(-1{)}^{i+j}{K}_{ij};$$

3. localized:

   $$E_{\mathrm{loc}}=\frac{1}{2}{K}_{jj}=G{m}^2\left[\frac{1}{\sqrt{\pi }a}-\frac{\mathrm{erf}(\text{abs}u/(2a))}{\text{abs}u}\right],$$

   with value zero at $u=0$, and

   $$E_{\mathrm{loc}}=\frac{G{m}^2\text{abs}{u}^2}{12\sqrt{\pi }{a}^3}+O(G{m}^2\text{abs}{u}^4/a^5);$$

4. domain-correlated:

   $${K^{\mathrm{D}}}_{cd}=\sum _{i\in D_c,j\in D_d}{K}_{ij},E_{\mathrm{D}}=\frac{1}{2}{\xi }^{\mathsf{T}}{K}^{\mathrm{D}}\xi .$$

   For independent symmetric domain signs,

   $$\mathbb{E}{E}_{\mathrm{D}}=\frac{1}{2}\sum _c{K}_{cc}^{\mathrm{D}},\mathrm{Var}(E_{\mathrm{D}})=\sum _{c<d}({K^{\mathrm{D}}}_{cd}{)}^2.$$

Proof. Expand the four Gaussian pairings. For every $s$, $s^{\mathsf{T}}Ks=G\text{norm}{{\sum }_i{s}_i{f}_i}_{\mathrm{C}}^2\ge 0$. The localized expansion follows from the Taylor series of $\mathrm{erf}$. For domains,

Distinct unordered sign products are orthogonal in expectation.