A Double--Scaling Large--$d$ Saddle of BFSS/BMN Matrix Quantum Mechanics

We study the large–$d$ dynamics of the mass–deformed bosonic ${\mathrm{B}\mathrm{F}\mathrm{S}\mathrm{S}}_{d+1}$ matrix quantum mechanics using a Hubbard–Stratonovich localization of the Yang–Mills interaction. After integrating out the matrix coordinates, the theory reduces to a holonomy–dependent effective action for an auxiliary adjoint kernel. We introduce a commuting–symmetric saddle and its maximally symmetric specialization, in which the interaction is encoded in a single dynamically generated mass shift $k_0$. The resulting large–$d$ description is a gauged matrix harmonic oscillator with self–consistent frequency $s^2=m+k_0$, fixed by a gap equation. We analyze the low–temperature $X$-space physics, the holonomy effective action, the Yang–Mills observable, and the associated phase structure. We then identify a correlated double–scaling limit in which $d\to \mathrm{\infty }$, $m\to \mathrm{\infty }$, and $\kappa =m^{3\mathrm{/}2}\mathrm{/}d$ is held fixed. In this limit the Yang–Mills interaction and the explicit mass deformation remain parametrically balanced: the theory interpolates between the commutator–dominated BFSS regime and the mass–dominated Gaussian regime. The double–scaled theory exhibits two complementary large–$d$ regimes. At low temperature, the enhanced gap pushes the deconfinement scale upward and opens a parametrically large uniform–holonomy region, where the bulk dynamics behaves as weakly coupled ${\mathrm{B}\mathrm{F}\mathrm{S}\mathrm{S}}_2$–type gauged harmonic–oscillator sectors. At the same time, the high–temperature branch reveals an overlap window in which the Gaussian description remains self–consistent while the commutator contribution per matrix pair is parametrically suppressed. The resulting dynamics is therefore ${\mathrm{B}\mathrm{F}\mathrm{S}\mathrm{S}}_2$–like in its enlarged uniform–holonomy sector and IKKT–like in its almost–commuting matrix behavior.