Gram--Wishart--Stiefel formulation of the $N=2$, large--$d$ gauge theory in 1D

We develop in this paper the Gram/Wishart/Stiefel formulation of the $N=2$, large–$d$ planar endpoint theory of the BFSS/BMN matrix quantum mechanics on the lattice, obtained in our previous work. In this formulation, the endpoint degrees of freedom are reorganized into rank–two Wishart eigenvalues and relative Stiefel angular variables. This allows the holonomy invariants $A$, $B$, and $R^2=A^2+B^2$ to be analyzed directly in terms of radial and angular Gram data. A central point is the large-$R$ aligned asymptotics of the holonomy potential. Its universal linear contribution $-A$ is absorbed into the Gaussian sector, producing the shifted mass parameter $({\alpha }_{\mathrm{\Lambda }}{)}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\alpha }_{\mathrm{\Lambda }}-1\mathrm{/}2$. In the Gram/Wishart/Stiefel variables, the exact $O(2)$ angular integral encodes this shifted sector in a rank–two Bessel kernel. The pure $-A$ theory, which is exactly solvable in Cartesian variables, then fixes the leading Bessel/HCIZ structure: its exponential part selects the aligned configuration, while its prefactor removes the spurious doubled Wishart entropy. We then apply this structure to the transverse $B$-type expansion and its non-polynomial toy completion. Finite polynomial truncations lead to an apparent large–$d$ perturbativity bound incompatible with the continuum limit, but this bound is shown to be an artifact of truncation. After summing the local transverse completion and balancing the compensating $+A$ term, the Wishart saddle is recovered with the physical shifted mass. The resulting continuum behavior reproduces the universal $-2d$ contribution of the $D_{\mathrm{\Lambda }}$-channel, while the genuinely anisotropic ${\beta }_{\mathrm{\Lambda }}$-channel lies outside the scope of a pure transverse $B$-type description.