The Galactic Center Excess (GCE) is an extended gamma-ray source in the central region of the Galaxy found in Fermi Large Area Telescope (Fermi-LAT) data. One of the leading explanations for the GCE is an unresolved population of millisecond pulsars (MSPs) in the Galactic bulge. Due to differing star formation histories it is expected that the MSPs in the Galactic bulge are older and therefore dimmer than those in the Galactic disk. Additionally, correlations between the spectral parameters of the MSPs and the spin-down rate of the corresponding neutron stars have been observed. This implies that the bulge MSPs may be spectrally different from the disk MSPs. We perform detailed modelling of the MSPs from formation until observation. Although we confirm the correlations, we do not find they are sufficiently large to significantly differentiate the spectra of the bulge MSPs and disk MSPs when the uncertainties are accounted for. Our results demonstrate that the population of MSPs that can explain the gamma-ray signal from the resolved MSPs in the Galactic disk and the unresolved MSPs in the boxy bulge and nuclear bulge can consistently be described as arising from a common evolutionary trajectory for some subset of astrophysical sources common to all these different environments. We do not require that there is anything unusual about inner Galaxy MSPs to explain the GCE. Additionally, we use a more accurate geometry for the distribution of bulge MSPs and incorporate dispersion measure estimates of the MSPs distances. We find that the elongated boxy bulge morphology means that some the bulge MSPs are closer to us and so easier to resolve. We identify three resolved MSPs that have significant probabilities of belonging to the bulge population.
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