  The central kiloparsecs of the Milky Way are known to host an old, spheroidal
stellar population, whose spatial and kinematical properties set it apart from
the boxy/peanut structure that constitutes most of the central stellar mass.
The nature of this spheroidal population, whether a small classical bulge, the
innermost stellar halo or a population of disk stars with large initial
velocity dispersion, remains unclear. This structure is also a promising
candidate to host some of the oldest stars in the Galaxy. Here we address the
topic of the inner stellar spheroid age, using spectroscopic and photometric
metallicities for a sample of 935 RR Lyrae stars that are constituents of this
component. By means of stellar population synthesis, we derive an
age-metallicity relation for RR Lyrae populations. We infer, for the RR Lyrae
stars in the bulge spheroid, an extremely ancient age of $13.41 \pm 0.54$ Gyr
and conclude they were among the first stars to form in what is now the Milky
Way galaxy. Our age estimate for the central spheroid shows remarkable
agreement with the age profile that has been inferred for the Milky Way stellar
halo, suggesting a connection between the two structures. However, we find mild
evidence for a transition in the halo properties at $r_{\rm GC} \sim 5$~kpc. We
also investigate formation scenarios for metal-rich RR Lyrae stars, such as
binarity and helium variations, and whether they can provide alternative
explanations for the properties of our sample. We conclude that, within our
framework, the only viable alternative is to have younger, slightly
helium-rich, RR Lyrae stars, a hypothesis that would open intriguing questions
for the formation of the inner stellar spheroid.
