\section{Proof of Lemma~\ref{lem:lift}}\label{app:lift}

\begin{proof}
Conjugating by $e^{-\pi\ii x/\varpi}$ turns the antiperiodic difference operator
$\tau_\varpi+I$ into a nonzero multiple of the ordinary difference operator
$\tau_\varpi-I$, so it suffices to treat $(\tau_\varpi-I)^{\Delta+1}U=0$ on the
$\varpi$-periodic open set $\R\setminus\Pi$. There, the distribution
$Y=(\tau_\varpi-I)^\Delta U/(\Delta!\,\varpi^\Delta)$ is $\varpi$-periodic.
Choose $\chi_\varpi\in C_c^\infty$ with $\sum_{n\in\Z}\chi_\varpi(x+n\varpi)=1$, and let
$v_\Delta$ be the periodization $\sum_n\tau_{n\varpi}$ of
$\chi_\varpi\,(\tau_\varpi-I)^{\Delta}U/(\Delta!\,\varpi^\Delta)$. It is a $\varpi$-periodic
tempered distribution which agrees with $Y$ on $\R\setminus\Pi$. Since
$(\tau_\varpi-I)^\Delta(x^\Delta v)=\Delta!\,\varpi^\Delta v$ for $\varpi$-periodic $v$,
the distribution $U-x^\Delta v_\Delta$ satisfies
$(\tau_\varpi-I)^{\Delta}(U-x^\Delta v_\Delta)=0$ on $\R\setminus\Pi$. We descend in the
degree, undo the conjugation, and put $W=U-\sum_rx^rv_r$, which vanishes on
$\R\setminus\Pi$.

All operations used are translations by at most $24p$, multiplication by a
compactly supported smooth function, periodization, modulation, and
multiplication by $x^r$. The transpose of periodization is
$\varphi\mapsto\chi_\varpi(x)\sum_n\varphi(x+n\varpi)$, whose local derivatives are
bounded by Schwartz seminorms with weight $2$, uniformly for
$\varpi\in[\frac12,2]$; by the norm equivalence of Section~\ref{sec:prelim},
periodization loses at most three Hermite orders, and multiplication by $x^r$
loses at most $r+1$. The total loss is at most
$6p+\sum_{r=0}^{12p}(r+4)=72p^2+60p+4<m_p$. The constants are independent of $\Pi$
and of the spacing of its points; in particular no division by a flat
annihilating function occurs.
\end{proof}