The bulge is the oldest component of the Milky Way. Since numerous simulations of Milky Way formation have predicted that the oldest stars at a given metallicity are found on tightly bound orbits, the Galaxy's oldest stars are likely metal-poor stars in the inner bulge with small apocenters (i.e., R apo ≲ 4 kpc). In the past, stars with these properties have been impossible to find due to extreme reddening and extinction along the line of sight to the inner bulge. We have used the mid-infrared metal-poor star selection of Schlaufman & Casey (2014) on Spitzer/Galactic Legacy Infrared Mid-Plane Survey Extraordinaire data to overcome these problems and target candidate inner bulge metal-poor giants for moderate-resolution spectroscopy with Anglo-Australian Telescope/AAOmega. We used those data to select three confirmed metal-poor giants ([Fe/H] =-3.15,-2.56,-2.03) for follow-up high-resolution Magellan/Magellan Inamori Kyocera Echelle spectroscopy. A comprehensive orbit analysis using Gaia DR2 astrometry and our measured radial velocities confirms that these stars are tightly bound inner bulge stars. We determine the elemental abundances of each star and find high titanium and iron-peak abundances relative to iron in our most metal-poor star. We propose that the distinct abundance signature we detect is a product of nucleosynthesis in the Chandrasekhar-mass thermonuclear supernova of a CO white dwarf accreting from a helium star with a delay time of about 10 Myr. Even though chemical evolution is expected to occur quickly in the bulge, the intense star formation in the core of the nascent Milky Way was apparently able to produce at least one Chandrasekhar-mass thermonuclear supernova progenitor before chemical evolution advanced beyond .