Molien--Weyl Singlet Counting and BFSS$_2$--Factorization in Gaussian Matrix QM

Abstract: We study the singlet-sector structure of mass-deformed BFSS$_{d+1}$ matrix quantum mechanics by combining the large--(d) Gaussian reduction with the Molien--Weyl projection. The Gaussian reduction captures the bulk matrix dynamics through a gauged harmonic oscillator, while the Molien--Weyl integral imposes the Gauss law and reorganizes the physical Hilbert space into holonomy-projected singlet excitations. We show that the very-low-temperature bosonic singlet spectrum is universally controlled by the quadratic Gram operators (\Tr(X_aX_b)), whose number is (d(d+1)/2). For (N=2), this result is established by explicit residue computations and character methods; for (N>2), it is supported by the character analysis. Thus the infrared spectrum begins as a collection of BFSS$_2$--like Gram towers, although higher invariant structures generally modify the full partition function. We also give a Hamiltonian derivation of the exceptional exact factorization at ((d,N)=(2,2)), where the BFSS$_3$ singlet partition function equals the cube of the BFSS$_2$ one for all temperatures. This rigidity is special to the (SU(2)) invariant tensor structure and explains why (d=1) and (N=2) are exceptional regimes without a deconfinement crossover. Finally, we extend the Gram-counting picture to supersymmetric BFSS/BMN models and indicate how the Molien--Weyl formulation can benchmark Monte Carlo simulations in both (X_a)-space and holonomy space.