\section{The positivity conjecture}\label{sec:positivity} \subsection{The statement and its hypothesis}\label{ssec:positivitystatement} A single-centred supersymmetric black hole in four dimensions has a spherically symmetric horizon and carries no angular momentum. The helicity trace index $B_6$ of a quarter-BPS state is $-\frac{1}{6!}\Tr\big[(-1)^{2h}(2h)^6\big]$, and for a configuration whose only degrees of freedom beyond the horizon are the twelve fermion zero modes of the broken supersymmetries, the trace evaluates to $-1$ times the number of horizon states; the sign is fixed and the index is negative, that is, $d = -B_6 > 0$. Sen \cite{Sen11positivity} turned this into a conjecture about Fourier coefficients. In the region $\mathcal R$ of the axion--dilaton moduli space in which the attractor point of the charges $(Q,P)$ lies, bounded by walls of marginal stability, the intended contribution is isolated under the exclusions described in the original argument, and the Fourier coefficient \eqref{eq:dvv} taken with the contour of that region should be positive. For the heterotic string on $T^6$ the region is cut out by \begin{equation}\label{eq:senconditions} \begin{gathered} Q\cdot P\ge 0,\qquad Q\cdot P\le Q^2,\qquad Q\cdot P\le P^2,\\ Q^2>0,\qquad P^2>0,\qquad Q^2P^2-(Q\cdot P)^2>0, \end{gathered} \end{equation} (equation (3.7) of \cite{Sen11positivity}), and for the $\ZZ_2$ CHL model, whose duality group is $\Gamma_1(2)$, by \begin{equation}\label{eq:z2conditions} \begin{gathered} Q^2>0,\qquad P^2>0,\qquad Q^2P^2>(Q\cdot P)^2,\\ 2Q^2+P^2\ge 3(Q\cdot P),\qquad 2Q^2\ge Q\cdot P,\\ P^2\ge Q\cdot P,\qquad Q\cdot P\ge 0 \end{gathered} \end{equation} (equation (3.9) there, with $Q^2$ normalised so that $NQ^2/2\in\ZZ$). The fourth condition is linear in $Q\cdot P$: it is the image of the wall $Q\cdot P\ge0$ under the second matrix of Sen's equation (3.8), $(Q,P)\mapsto(-Q+P,-2Q+P)$, whose inner product is $2Q^2-3\,Q\cdot P+P^2$. Equations (3.25) and (4.7) of \cite{CD19} print it as $2Q^2+P^2-3(Q\cdot P)^2\ge0$; that squared form is a misprint of the same condition, and it would exclude, for instance, the triple $(Q^2,P^2,Q\cdot P)=(3,2,2)$ of Table~\ref{tab:violations}. Sen verified the positivity of $d(Q,P)$ on these cones for the heterotic string and for the CHL models $N=2,3,5,7$ in the range of charges of his Tables~1 to~5, and stated the conjecture for all charges in the cone. We have reproduced his Tables~1 and~2 exactly (Appendix~\ref{app:computation}). The hypothesis of the argument is explicit in \cite{Sen11positivity}: the conclusion requires ``some duality frame in which only single centered black hole solution, whose only hair are the fermion zero modes associated with broken supersymmetry, contributes'' to the index. In the heterotic frame the quarter-BPS black hole does have hair beyond the fermion zero modes (the hair modes identified in \cite{BMS09hair}), but there is a frame, the type IIB frame on $K3\times T^2$, in which the hair reduces to the fermion zero modes; this is what makes the conjecture consistent there, and what was proved later. \begin{theorem}[Bringmann--Murthy 2013]\label{thm:bringmannmurthy} For type II string theory on $K3\times T^2$ (equivalently the heterotic string on $T^6$, the form $1/\Phi_{10}$) and on $T^6$ (the $\mathcal N=8$ theory), the single-centred indices in the attractor region are positive for the classes of charges treated in \cite{BringmannMurthy13}; the proof uses the mock modular decomposition of the Fourier--Jacobi coefficients of $1/\Phi_{10}$ \cite{DMZ12}. \end{theorem} The restriction to classes of charges was removed by Rossell\'o \cite{Rossello24}, who bounds the tail of the Rademacher expansion of the mock Jacobi forms $\psi^F_m$ of \cite{DMZ12} against its leading term. \begin{theorem}[Rossell\'o 2024]\label{thm:rossello} For type II string theory on $K3\times T^2$, the single-centred index $d^*(m,n,\ell)$, with $m = P^2/2$, $n = Q^2/2$, $\ell = Q\cdot P$, is positive for every $m,n\ge 1$ with $4mn-\ell^2>0$: for $n\ge m$ and $0\le \ell\le m$ one has $(-1)^{\ell+1}c^F_m(n,\ell)>0$, and the general case follows by the $S$-duality invariance of the single-centred index. \end{theorem} \begin{verdict}[claim 9, heterotic and CHL] Proved for the heterotic string on $T^6$, by Theorems~\ref{thm:bringmannmurthy} and~\ref{thm:rossello}; our scan of the single-centred cone up to $Q^2\le 12$, $P^2\le 6$, $Q\cdot P\le 4$ (Appendix~\ref{app:computation}) agrees. For the CHL orbifolds no counterexample is known, our scan for $N=2$ finds only positive indices, while a general CHL proof is not supplied here: the Rademacher bound of \cite{Rossello24} has not been carried out for the mock Jacobi forms of $1/\Phit_k$, and \cite{Rossello24} names this as the next case. The conjecture is proved in the model in which it was first stated and confirmed in the CHL models. \end{verdict} \subsection{Full and horizon indices in the toroidal orbifolds}\label{ssec:toroidal} Sen's paper states the conjecture as a consequence of the black hole argument for ``the $\mathcal N=4$ supersymmetric string theories'', and the subsequent literature \cite{CD19,CKMRW20} tested it in the full list of $\mathcal N=4$ compactifications with known dyon partition functions. Chattopadhyaya and David \cite{CD19} found that it fails in two of them: the freely acting $\ZZ_2$ and $\ZZ_3$ orbifolds of type IIB on $T^6$, in which the orbifold acts by a reflection of four coordinates of $T^4$ (or a rotation by $2\pi/3$ of two planes) together with a shift of half (or a third) of a circle of the remaining $T^2$. The $\ZZ_2$ model has the twisted elliptic genera \begin{equation}\label{eq:tor2genera} \begin{aligned} F^{(0,0)} &= 0,& F^{(0,1)} &= \tfrac83 A - \tfrac43 B\,E_2(\tau),\\ F^{(1,0)} &= \tfrac83 A + \tfrac23 B\,E_2(\tfrac\tau2),& F^{(1,1)} &= \tfrac83 A + \tfrac23 B\,E_2(\tfrac{\tau+1}2), \end{aligned} \end{equation} where $A = \sum_{i=2}^4\theta_i(\tau,z)^2/\theta_i(\tau,0)^2$, $B = \theta_1(\tau,z)^2/\eta(\tau)^6$, and $E_2(\tau) := 2E(2\tau) - E(\tau)$ is the weight-two form of $\Gamma_0(2)$ built from the Eisenstein series $E(\tau) = 1 - 24\sum\sigma_1(n)q^n$ (this is the $E_N$ of \cite{DJS06}; it is not the quasi-modular Eisenstein series, and the two are easily confused). The Siegel form is $\Phit_2 = \Phit_6^2/\Phit_{10}$, of weight two, with the prefactor $e^{2\pi i(\rho+v)}$ in place of $e^{2\pi i(\rho+\sigma/N+v)}$ in \eqref{eq:product}, and the index is given by \eqref{eq:dvv} with $Q^2\in\ZZ$, $P^2\in 2\ZZ$. The single-centred region is again cut out by \eqref{eq:z2conditions}, since the duality group is again $\Gamma_1(2)$. \begin{theorem}[known negative full indices; finite exact test]\label{thm:positivityfails} In the $\ZZ_2$ toroidal orbifold of type IIB on $T^6$, the index $d(Q,P)$ computed from \eqref{eq:dvv} with the contour of the attractor region is negative for infinitely many charges satisfying the single-centred conditions \eqref{eq:z2conditions}. In the range $Q^2\le 12$, $P^2\le 8$, $Q\cdot P\le 4$ there are exactly thirty such charge triples, listed in Table~\ref{tab:violations}; the first ones are \begin{gather*} d(1,2,0) = -224,\quad d(1,4,0) = -1248,\\ d(2,2,1) = -256,\quad d(3,2,0) = -3392,\\ d(3,4,0) = -10320, \end{gather*} in the notation $d(Q^2,P^2,Q\cdot P)$. In the first column of Table~\ref{tab:tor2} ($P^2=0$, outside the cone) the sign is $(-1)^{Q^2}$ for $Q^2\le 12$; Figure~\ref{fig:indices} draws the table. \end{theorem} \begin{figure}[tbp] \centering \includegraphics[width=0.94\linewidth,height=0.48\textheight,keepaspectratio]{figures/fig_indices.png} \caption{The full index $d(Q^2,P^2,0)$ of the $\ZZ_2$ toroidal orbifold from Table~\ref{tab:tor2}, drawn on the scale $\mathrm{sign}(d)\log_{10}(1+|d|)$. Every bar in burnt orange is a negative full index; along $P^2 = 2$ the sign alternates with $Q^2$, and every odd $Q^2$ lies inside the cone \eqref{eq:z2conditions} of Theorem~\ref{thm:positivityfails}. A horizon index requires the separate hair analysis.} \label{fig:indices} \end{figure} \begin{proof} We expanded the product \eqref{eq:product} for the genera \eqref{eq:tor2genera} in exact rational arithmetic, with the Fourier coefficients of the theta functions, of $\eta^{-6}$ and of the Eisenstein series computed from their definitions, and read off the coefficients in the convention $q_\rho=e^{2\pi i\rho}$, $q_\sigma=e^{2\pi i\sigma/N}$ and $y=e^{2\pi iv}$, keyed by $(NQ^2/2,P^2/2,Q\cdot P)$ in the expansion in which the factor $(1-e^{-2\pi i v})^{-2}$ at $k'=l=0$ is expanded in powers of $e^{-2\pi iv}$ (the attractor region for $Q\cdot P\ge 0$). The resulting tables agree entry by entry with Tables~17, 18 and~19 of \cite{CD19} in their common range. The thirty charge triples of Table~\ref{tab:violations} satisfy each inequality of \eqref{eq:z2conditions} by inspection and have negative index. For the infinitude: the values at $P^2=2$, $Q\cdot P=0$ are \[ \begin{gathered} 64,\ -224,\ 1152,\ -3392,\ 11520,\ -30336,\ 83968,\\ -202560,\ 496512,\ -1118496,\ 2521600,\ -5374656 \end{gathered} \] for $Q^2 = 0,1,2,\dots,11$; every odd $Q^2$ with $P^2 = 2$, $Q\cdot P = 0$ lies in the cone \eqref{eq:z2conditions}, and \cite[Section~4.1]{CD19} show from the Fourier--Jacobi coefficient $\psi_1$ of $1/\Phit_2$ at $P^2=2$ that the finite part of $\psi_1$, which is what the single-centred index is, has coefficients of alternating sign in $Q^2$; the alternation is visible in our table for $Q^2\le 12$. The analogous statement for the $\ZZ_3$ orbifold is \cite[Section~4.2]{CD19}. \end{proof} \begin{table}[tbp] \centering\scriptsize \begin{tabular}{@{}rrrr@{\quad}rrrr@{}} \toprule $Q^2$&$P^2$&$Q\cdot P$&$d$&$Q^2$&$P^2$&$Q\cdot P$&$d$\\ \midrule 1 & 2 & 0 & -224 & 6 & 2 & 1 & -33536 \\ 1 & 4 & 0 & -1248 & 7 & 2 & 0 & -202560 \\ 1 & 6 & 2 & -224 & 7 & 2 & 2 & -30336 \\ 1 & 8 & 2 & -1248 & 7 & 4 & 0 & -179022 \\ 2 & 2 & 1 & -256 & 7 & 4 & 4 & -10320 \\ 2 & 6 & 3 & -256 & 8 & 2 & 1 & -220672 \\ 3 & 2 & 0 & -3392 & 9 & 2 & 0 & -1118496 \\ 3 & 2 & 2 & -224 & 9 & 2 & 2 & -202560 \\ 3 & 4 & 0 & -10320 & 9 & 4 & 0 & -155232 \\ 3 & 6 & 4 & -224 & 9 & 4 & 4 & -55424 \\ 4 & 2 & 1 & -3840 & 10 & 2 & 1 & -1204992 \\ 5 & 2 & 0 & -30336 & 11 & 2 & 0 & -5374656 \\ 5 & 2 & 2 & -3392 & 11 & 2 & 2 & -1118496 \\ 5 & 4 & 0 & -55424 & 11 & 4 & 4 & -179022 \\ 5 & 4 & 4 & -1248 & 12 & 2 & 1 & -5741824 \\ \bottomrule \end{tabular} \caption{The thirty negative full-index triples in the linear cone \eqref{eq:z2conditions}, for $0