PomeLemma 11.23.14.6C85B622ADCF The averages of $|S|$ and $P|S|$ are
\[ \int _ X|S|\mu =\frac8{\pi ^2}, \qquad \int _ XP|S|\mu =\frac8{3\pi ^2}. \]
Lemma 11.23.146C85B622ADCF
Proof. The integrands do not depend on $y_1,y_2$. The degree-two map of real tori $(a,b)\mapsto (a+b,a-b)$ preserves normalized Haar averages. With the angles $A=2\pi a$ and $B=2\pi b$, it gives
\[ S=2\cos A\cos B,\qquad P=\frac{\cos (2A)+\cos (2B)}2. \]
The one-variable averages of $|\cos A|$ and $|\cos A|^3$ are $2/\pi $ and $4/(3\pi )$, respectively, so the average of $\cos (2A)|\cos A|$ is $2/(3\pi )$. Multiplying the averages over the two independent angles gives both formulas. $\square$
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