Dimension Monotonicity in Laguerre Ensembles II: Average Singular Values and the Rectangularity Transition in the Orthogonal Case

We study the normalized half moment $α_{\mathbb R}^{(λ)}(N)$ of the size-$N$ Laguerre orthogonal ensemble for real shape $λ\ge0$. At integer shape this is the expected average singular value of an $N\times(N+λ)$ real Gaussian matrix. The square mean increases with the dimension, whereas every real s