Molien–Weyl Singlet Counting and BFSS 2 -Factorization in Gaussian Matrix Quantum Mechanics

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 a X 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 distinguishes exact singlet factorization from the possibility of smooth deconfinement–like holonomy crossovers at finite N. 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.