Denominator-bounded rational phase reconstruction

Let $N,Q$ be integers with $N\ge 1$ and $Q>4N^2$. A measured phase label is represented by the unique integer $y$ with $0\le y<Q$ as in For $n\ge 1$, every class in $\mathbb{Z}\mathrm{/}n$ has one representative $r$ with $0\le r<n$, so $\mid \mathbb{Z}\mathrm{/}n\mid =n$; while $\mathbb{Z}\mathrm{/}0$ is in bijection with $\mathbb{Z}$. Write the normalized finite regular continued fraction of $y\mathrm{/}Q$ as $[a_0;\dots ,a_s]$ and denote its convergents by $p_j\mathrm{/}{q}_j$. Every convergent is reduced by Determinant identity for consecutive convergents. A candidate is a convergent whose numerator and denominator are reduced, whose denominator satisfies $1\le q_j\le N$, and which passes the exact integer test $2\mathrm{\mid }y{q}_j-p_{j}Q\mathrm{\mid }\le q_j\mathrm{.}$

The reconstruction result is the unique candidate if the candidate set is nonempty, and is FAIL otherwise. The candidate set includes the terminal convergent whenever it meets the same conditions. Uniqueness, and hence well-definedness of this value, follows from Uniqueness of denominator-bounded phase candidates under $Q>4N^2$.