JADES and BlackTHUNDER : rest-frame Balmer-line absorption and the local environment in a Little Red Dot at z = 5

We present a 'Little Red Dot' (LRD) broad-line active galactic nucleus (AGN) at z = 5.077 observed with NIRSpec/MSA (micro-shutter assembly) and NIRSpec/IFU (integral-field unit) by the JADES (JWST Advanced Deep Extragalactic Survey) and BlackTHUNDER (Black holes in THe early Universe aNd their DensE surRoundings) surveys. Combining spatially resolved and high-resolution spectroscopy, we characterize its central engine, host, and environment. Ho has multiple components, including two broad Gaussians, yielding a black-hole mass log(M-center dot/M-circle dot) = 7.65, while [O III]lambda 5007 gives a galaxy dynamical mass log(M-dyn/M-circle dot) 9.1, suggesting an overmassive black hole relative to the host galaxy. The target is immersed in a 7-kpc wide pool of ionized gas and has three neighbours: a satellite galaxy, a possible satellite/gas cloud, and a tentatively detected spatially detached outflow. Hoa shows strong, rest-frame absorption, deeper than the continuum, ruling out a stellar origin. The velocity and velocity dispersion are V-ahs-13 km s(-1) and sigma(abs) = 120 km s(-1). There is tentative evidence (2.6 sigma) of temporal variability in the equivalent width of the Hot absorber over two rest-frame months, suggesting a highly dynamic nucleus. Notably, while the Ha absorber is clearly visible and even dominant in the high-resolution G395H observations, it is not detected in the medium-resolution G395M data of the same epoch. This implies that the current incidence rate of absorbers in LRDs - and especially of rest-frame absorbers - may be severely underestimated, because most LRDs rely on lower resolution spectroscopy. The high incidence rate of rest-frame absorbers in LRDs may indicate a configuration that is either intrinsically stationary, such as a rotating disc, or that exhibits time-averaged stability, such as an oscillatory "breathing mode' accretion with cyclic expansion and contraction of the gas around the supermassive black hole.