  The formation of supermassive black holes by direct collapse would imply the
existence of supermassive stars (SMSs) and their collapse through the
general-relativistic (GR) instability into massive black hole seeds. However,
the final mass of SMSs is weakly constrained by existing models, in spite of
the importance this quantity plays in the consistency of the direct collapse
scenario. We estimate the final masses of spherical SMSs in the whole parameter
space relevant for these objects. We build analytical stellar structures
(hylotropes) that mimic existing numerical SMS models accounting for full
stellar evolution with rapid accretion. From these hydrostatic structures, we
determine ab initio the conditions for GR instability, and compare the results
with the predictions of full stellar evolution. We show that hylotropic models
predict the onset of GR instability with high precision. The mass of the
convective core appears as a decisive quantity. The lower it is, the larger is
the total mass required for GR instability. Typical conditions for GR
instability are a total mass >10^5 Msun with a core mass >10^4 Msun. If the
core mass remains below 10^4 Msun, total masses in excess of 10^6-10^7 Msun can
be reached. Our results confirm that spherical SMSs forming in primordial,
atomically cooled haloes collapse at masses below 500 000 Msun. On the other
hand, accretion rates in excess of 1000 Msun/yr, leading to final stellar
masses >10^6 Msun, are required for massive black hole formation in metal-rich
gas. Thus, the different channels of direct collapse imply distinct final
masses for the progenitor of the black hole seed.
