How I Wonder What You Are -- JWST's Little Red Dots do not TWINKLE

Little Red Dots (LRDs) are a population of compact, red sources that have emerged as one of the most puzzling findings of JWST. Variability provides a direct probe of their central engines. Here we present the first joint spectroscopic and photometric time-domain study of LRDs undertaken with the JWST TWINKLE slitless spectroscopy program. Surveying the FRESCO GOODS-North legacy field, TWINKLE monitors a complete, H$α$-flux-limited sample of 18 LRDs at z = 3.9-6.8, achieving a rest-frame baseline of $\sim$140-220 days. We detect no variability in photometry, H$α$ line flux, or line shape across the sample. If LRDs resembled AGN in reverberation mapping samples -- the foundation for black hole mass calibrations and luminosity scaling relations -- we would expect >10 sources to show measurable fluctuations. Observing none implies a 5.9$σ$ deficit. The non-detections hold across all broad H$α$ emitters within TWINKLE's field of view -- the 18 V-shaped LRDs as well as 9 non-LRDs. Comparison with simulated light curves disfavors sub-Eddington accretion and is instead consistent with super-Eddington accretion, other mechanisms that suppress variability, or perhaps no AGN whatsoever. If LRDs do harbor black holes, calibrations derived from sub-Eddington systems may not apply, thereby explaining JWST's apparently "overmassive" black holes. These observations provide unique constraints on the physics of one of the most enigmatic populations discovered by JWST.