Lower Bounds for Admissible Functions on $\Lambda_n$

We prove an explicit, quantitative lower bound for all $g$-admissible functions on the Lagrangian Grassmannian $\Lambda_n=U(n)/O(n)$ that are invariant under the maximal torus $\mathbb{T}^{n-1}\subset U(n)$ and have zero supremum. The method of Ben Abdesselem--Jelloul, developed for the complex Grassmannian $G_{2,4}\mathbb{C}$, cannot be transposed directly because (i) the holomorphic isometry group of $\Lambda_n$ is too small to reduce the problem to diagonal matrices, and (ii) there is no holomorphic curve contained in $\Lambda_n$ through the relevant orbits. We resolve (i) by a torus reduction to radial functions in the affine cell $U_0\cong\mathbb{C}^{n-1}$, and (ii) by replacing the complex maximum principle with a real maximum principle along the diagonal radial curve. This yields an explicit scalar ODE whose integration gives a positive, quantitative lower bound, from which we deduce the integrability $\int_{\Lambda_n}e^{-\alpha\varphi}\,dv<\infty$ for $\alpha<\min(1/2,1/(n-1))$. We do \emph{not} claim the optimal Tian constant $1/(n+1)$; we explain precisely why the method falls short of it and what would be needed to close the gap. All computations are written out in full.