Exact zero-label probability for counting

For a phase $\varphi$ and label zero, the kernel in [F2] and geometric-sum fact [F4] give $L^{-2}\mathrm{\mid }{\sum }_{j=0}^{L-1}{e}^{2\pi ij\varphi }{\mathrm{\mid }}^2={\sin }^2(\pi L\varphi )\mathrm{/}(L^2{\sin }^2(\pi \varphi ))$ when $\varphi \mathrm{\notin }\mathbb{Z}$, by Euler’s formula. At an integer phase all summands are $1$, so the probability is one.

For $0<t<N$, [F1]–[F3] average this expression over $\varphi =\theta \mathrm{/}\pi$ and $1-\theta \mathrm{/}\pi$ with weights $1\mathrm{/}2$. For the second branch, [F8] gives $\sin (\pi -\theta )=\sin \theta$ and, by iterating the $\pi$-shift, $\sin (L\pi -L\theta )=(-1{)}^{L+1}\sin (L\theta )$ for the integer $L$. Squaring these identities shows that both branches have the same squared numerator and denominator, so their mixture is the stated value. For $t=0$, the phase is zero and label zero is certain.

When $L=2$, [F5] gives $\sin (2\theta )=2\sin \theta \cos \theta$, so the quotient in step 2.1 is ${\cos }^2\theta$. By [F6], ${\sin }^2(\pi y\mathrm{/}L)=0$ for an integer label $0\le y<L$ exactly when $y$ is a multiple of $L$, hence exactly when $y=0$. Fact [F7] identifies this with a zero raw count estimate.