Little Red Dots host Black Hole Stars: A unified family of gas-reddened AGN revealed by JWST/NIRSpec spectroscopy

We use the DAWN JWST Archive to construct and characterise a sample of 146 little red dots (LRDs) across 2.0<z<9.3, selecting all sources with v-shaped UV-optical continua from NIRSpec/PRISM spectra and compact morphologies in NIRCam/F444W imaging. We show that LRD continuum spectra are ubiquitously well described by modified blackbodies across ~$0.4-1.0\mu$m, with typical T~5000K or $\lambda_{peak}$~$0.65\mu$m across 2 dex in luminosity, and a tail toward T~2000K. LRDs therefore trace a locus in the Hertzsprung-Russell diagram that is directly analogous to stars on the Hayashi track, strongly supporting the picture that LRDs are AGN embedded in optically-thick dense gas envelopes. Hotter LRDs with $\lambda_{peak}<0.65\mu$m typically have strong Balmer breaks, redder UV slopes and high optical luminosities; other LRDs show weak or no Balmer breaks, and wide variety in $\beta_{UV}$ and $L_{5100}$. Crucially, we demonstrate that the UV-optical continuum shapes and luminosities are strongly linked to the $H\alpha,\ H\beta$, [OIII] and OI line properties. There is a tight linear relation between the H$\alpha$ and optical continuum luminosities, as well as H$\alpha$ and OI$_{8446}$, indicating that Balmer, OI and optical emission must primarily be powered by the same source. The Balmer decrement increases strongly toward higher $L_{H\alpha}$, $L_{5100}$ and Balmer break strength, providing key evidence for luminosity-dependent effects of collisional (de-)excitation and resonant scattering in the gaseous envelopes. In contrast, we show that [OIII] emission likely originates from star-forming host galaxies, and that its strong correlation with Balmer break strength arises naturally from variation in the AGN-to-host ratio among the LRD population. Our work presents an empirical description of the nature and structure of LRDs, defining a new benchmark for ongoing LRD model developments.