Data for the paper titled "The Missing Hard Photons of Little Red Dots: Their Incident Ionizing Spectra Resemble Massive Stars"

The nature of Little Red Dots (LRDs) has largely been investigated through their continuum emission, with lines assumed to arise from a broad-line region. In this paper, we instead use recombination lines to infer the intrinsic properties of the central engine of LRDs. Our analysis first reveals a tension between the ionizing properties implied from H$\alpha$ and He\,\textsc{ii}$\,\lambda$4686. The high H$\alpha$ EWs require copious H-ionizing photons, more than the bluest AGN ionizing spectra can provide. In contrast, He\,\textsc{ii} emission is marginally detected, and its low EW is, at most, consistent with the softest AGN spectra. The low He\,\textsc{ii}/H$\beta$ ($\sim10^{-2}$, $<20\times$ local AGN median) further points to an unusually soft ionizing spectrum. We extend our analysis to consider dense gas envelopes (the ``quasi-star'' or ``black-hole star'' hypotheses), and find that hydrogen recombination lines become optically thick and lose diagnostic power, but He\,\textsc{ii} remains optically thin and a robust tracer. Photoionization modeling with \texttt{Cloudy} rules out standard AGN accretion disk spectra. Alternative explanations include: exotic AGN with red rest-optical emission; a very high {\it average} optical depth ($>10$) from gas/dust; and/or soft ionizing spectra with abundant H-ionizing photons, consistent with e.g., a cold accretion disk or a composite of AGN and stars. The latter is an intriguing scenario since high hydrogen densities are highly conducive for star formation, and nuclear star clusters are found in the immediate vicinity of local massive black holes. While previous studies have mostly focused on features dominated by the absorbing hydrogen cloud, the HeII-based diagnostic proposed here represents a crucial step toward understanding the central engine of LRDs.