Intermediate-mass black holes (IMBHs) of mass M • ≈ 10 2 -10 5 solar masses, M , are the long-sought missing link 1 between stellar black holes, born of supernovae 2 , and massive black holes 3 , tied to galaxy evolution by the empirical M • /σ correlation 4,5 . We show that low-mass black hole seeds that accrete stars from locally dense environments in galaxies following a universal M • /σ relation 6, 7 grow over the age of the Universe to be above M 0 ≈ 3 × 10 5 M (5% lower limit), independent of the unknown seed masses and formation processes. The mass M 0 depends weakly on the uncertain formation redshift, and sets a universal minimal mass scale for present-day black holes. This can explain why no IMBHs have yet been found 3 , and it implies that present-day galaxies with σ < S 0 ≈ 40 km s −1 lack a central black hole, or formed it only recently. A dearth of IMBHs at low redshifts has observable implications for tidal disruptions 8 and gravitational wave mergers 9 . The early stages of massive black hole growth are poorly understood 10 . High-luminosity active galactic nuclei at very high redshift 11 z further imply rapid growth soon after the Big Bang. Suggested formation mechanisms typically rely on the extreme conditions found in the early Universe (very low metallicity, very high gas or star density). It is therefore plausible that these black hole seeds were formed in dense environments, at least a Hubble time ago (z > 1.8 for a look-back time of t H = 10 Gyr) 12 .β between black hole mass, M • , and stellar velocity dispersion, σ , that is observed in the local Universe over more than about three decades in massive black hole mass, correlates M • and σ on scales that are well outside the massive black hole's radius of dynamical influence 3 ,2 . Recent analyses of large heterogeneous galaxy samples find that a universal M • /σ relation holds for all galaxy types 6, 7 , although the scope of this relation and its evolution with redshift remain controversial 13 . Here we adopt the empirical fit ), where δ = 0.49 ± 0.03 is the root mean square of the intrinsic scatter. We assume that this universal M • /σ holds at all redshifts 14 , and that the black hole seeds grow in a locally (within a few r h ) dense stellar environment. By fixing r h , the M • /σ relation then imposes tight connections between the black hole and the dynamical properties of its stellar surroundings 15 , and specifically the rate at which it consumes stars (see Methods section).A central black hole grows by (1) the accretion of stars, compact remnants and dark matter particles that are deflected toward it on nearly radial orbits, and either fall whole through the event horizon or are tidally disrupted outside it, and then accreted; (2) viscosity-driven accretion of interstellar gas; and (3) mergers with other black holes. Of these growth channels, only the accretion of stars must follow from the existence of a central black hole in a stellar system. Moreover, the tidal disruption event * Department of Particle Physics & Astrophysics, W...