3)$ and the claim follows, with with $\delta = (1/2)\delta'.$

Theorem \[LAPLA\] is proved in Section \[PROLAPLA\] by Laplace method.

\[LAPLA\] Let $U={\cal U}_\varepsilon(1/3, 1/3).$ There is an $\varepsilon>0$ such that\
$\sum_{\bar{\lambda}  , \bar{\omega} \in U} N(\bar{w}, \bar{l})/N_0 \,< \, C \cdot 3^{(1-\gamma)n}.$\

[*Proof of Theorem \[EX2EI\].*]{} Pick $\varepsilon $ such that Theorem \[LAPLA\] applies. Use Corollary \[SUMOPT\], Corollary \[SUMOMEGA\], and Theorem \[LAPLA\] and the sum of all terms $N(\bar{w}, \bar{l})/N_0$ with $l_i>0$ is $\le C \cdot 3^{(1- \gamma)n}.$ Terms with an $l_i=0$ do not add substantially to the sum (proof omittted.)

Proof of Theorem \[OPT\] {#PROOPT}
========================

We use the notation $\bar{x}= (x_0,x_1, x_2), \bar{y}=(y_0, y_1, y_2)$ and define $$\begin{aligned}
\mbox{OPT}_1(\bar{x} ,  s)\,= \,  \frac{ q(sx_0)}{q(s)}  \,+ \, \frac{q(sx_1)}{q(s)} + \frac{ q(sx_2)}{q(s)} \nonumber \\
\mbox{OPT}_2(\bar{x}, \bar{y} ,  s)\,= \, \left( \frac{1}{x_0y_0+x_1y_1+ x_2y_2} \right)^{Q}  , \, x_0y_0+x_1y_1+ x_2 y_2 >0\nonumber \\
 \mbox{OPT}_3(\bar{y}, s)\,= \, \left( y_0+y_1+y_2 \right )^Q \,+ \, 
2\cdot \left( y_0^2 \,+\,y_1^2\,+\, y_2^2\,- y_0 y_1\,- \, y_0 y_2\, - \, y_1 y_2 \right)^{1/2\cdot Q} \nonumber \\
\mbox{OPT}(\bar{x}, \bar{y},  s) \,=\, 
\mbox{OPT}_1(\bar{x} , s) \cdot 
\mbox{OPT}_2 ( \bar{x}, \bar{y} ,  s)\cdot 
\mbox{OPT}_3(  \bar{y}, s) .  \nonumber \end{aligned}$$ Observe that OPT$(1, 1, 1, 1, 1,1 , s)= 3(1/3)^Q 3^Q=3=$ OPT$_1(1, 