\section{The computation of the Fourier coefficients}\label{app:computation}

The tables of this paper are Fourier coefficients of the inverse of the product \eqref{eq:product}, computed in exact rational arithmetic. We record the steps so that the numbers can be reproduced without the program, which is in the archive \cite{Bha26}.

\paragraphx{The genera.} The twisted elliptic genera are expanded in $q = e^{2\pi i\tau}$ and $y = e^{2\pi i z}$ from the theta functions $\theta_i(\tau,z)$ and $\eta(\tau)$ as $q$-series with coefficients in $\ZZ[y,y^{-1}]$, and the coefficients $c^{(r,s)}_b(4n-j^2)$ are read off from $\sum_{n,j} c^{(r,s)}_b(4n-j^2)\,q^n y^j$. For K3, $F^{(0,0)} = 8A$ with $A = \sum_{i=2}^4\theta_i(\tau,z)^2/\theta_i(\tau,0)^2$; for the $\ZZ_2$ CHL model the genera are $F^{(0,0)} = 4A$, $F^{(0,1)} = \tfrac43 A - \tfrac23 B\,E_2(\tau)$, $F^{(1,0)} = \tfrac43 A + \tfrac13 B\,E_2(\tau/2)$, $F^{(1,1)} = \tfrac43 A + \tfrac13 B\,E_2((\tau+1)/2)$, with $B = \theta_1(\tau,z)^2/\eta(\tau)^6$ and $E_2(\tau) = 2E(2\tau)-E(\tau)$, where $E$ is the Eisenstein series of weight two normalised to $1 - 24\sum_{n\ge1}\sigma_1(n)q^n$; for the $\ZZ_2$ toroidal orbifold they are \eqref{eq:tor2genera}. The arguments $\tau/2$ and $(\tau+1)/2$ are handled by expanding in $q^{1/2}$ with the sign $(-1)^n$ on the odd powers, so every series is a Laurent series in $x = q^{1/8}$, which is the smallest fractional power that occurs.

\paragraphx{The product.} Each factor $(1 - e^{2\pi i(k'\sigma + l\rho + jv)})^{-c}$ of $1/\Phit$ is expanded as a binomial series in the monomial $e^{2\pi i(k'\sigma + l\rho + jv)}$ to the order needed, and the factors are multiplied as polynomials in the three variables $e^{2\pi i\rho}$, $e^{2\pi i\sigma/N}$, $e^{2\pi iv}$ truncated at the exponents $Q^2/2\le L$, $NP^2/2\le K$ wanted in the tables. The exponent $c = \sum_s e^{2\pi i s l/N}c^{(r,s)}_b(4k'l-j^2)$ is an integer for $N\le 2$, where the root of unity is $\pm1$; for $N=3,5,7$ it would require cyclotomic arithmetic, and we have not computed those models. The one factor with $k'=l=0$, $(1 - e^{\mp 2\pi iv})^{-2}$ after the prefactor is taken into account, is the only one whose expansion depends on the chamber: in the chamber $\operatorname{Im} v<0$ it is expanded in powers of $e^{-2\pi iv}$, in the chamber $\operatorname{Im} v>0$ in powers of $e^{2\pi iv}$. The two expansions give the two columns of Table~\ref{tab:phi10} at $\pm Q\cdot P$, and their difference is the wall-crossing jump of Theorem~\ref{thm:wallcrossing}, which the program checks for every $Q^2,P^2\in\{2,4,6\}$ and $|Q\cdot P|\le 4$.

\paragraphx{The index.} The coefficient of $e^{2\pi i(\rho Q^2/2 + \sigma P^2/2 + v\,Q\cdot P)}$ in $1/\Phit$, multiplied by $(-1)^{Q\cdot P+1}$, is the index $d(Q,P)$ of \eqref{eq:dvv}. For the orbifolds the prefactor of the product is $e^{2\pi i(\rho + \sigma/N + v)}$ for the CHL models and $e^{2\pi i(\rho + v)}$ for the toroidal orbifolds, as in \cite{CD19}, and the integral over $\sigma$ runs over a period of length $N$ (Remark~\ref{rem:normalisation}).

\paragraphx{The tables.} Table~\ref{tab:phi10} is the heterotic index for $Q^2,P^2\in\{2,4,6\}$ and $-2\le Q\cdot P\le 4$; it reproduces Table~1 of \cite{Sen11positivity} entry by entry. Table~\ref{tab:tor2} is the index of the $\ZZ_2$ toroidal orbifold for $Q^2\le 12$, $P^2\in\{0,2,4\}$ and $Q\cdot P\in\{0,1,2\}$; it agrees with Tables~17 to~19 of \cite{CD19} in every entry with $Q^2\le 12$ but one, the entry $Q^2=4$, $P^2=2$, $Q\cdot P=0$, which \cite{CD19} prints as $-11\,520$ and which the product formula gives as $+11\,520$, the sign required by the rule $(-1)^{Q^2}$ of \cite{CD19} for $P^2=2$; the misprint lies inside the single-centred cone and would otherwise be a thirty-first negative entry. Table~\ref{tab:violations} lists the charges in the cone \eqref{eq:z2conditions} with $Q^2\le 12$, $P^2\le 8$, $Q\cdot P\le 4$ whose index is negative. The same scan over the cone of the heterotic string and of the $\ZZ_2$ CHL model, up to $Q^2\le 12$ (heterotic) or $NQ^2/2\le 6$ (CHL), $P^2\le 6$, $Q\cdot P\le 4$, finds no non-positive index, and the CHL values reproduce Table~2 of \cite{Sen11positivity}.

\input{tables/phi10}
\input{tables/tor2}