Power-kernel fractional Sturm--Liouville operators: a graph realization and boundary-independent singular-value asymptotics

For $\frac{1}{2}<\alpha <1$ we construct a closed realization of $\mathcal{L}u={\mathcal{D}}_1^{\alpha }{D}_0^{\alpha }u$ in $L^2(0,1)$ whose domain contains the singular mode $x^{\alpha -1}$. The domain, defined via the distributional Caputo composition, is characterized by $I_0^{1-\alpha }u\in H^1(0,1)$ and $D_0^{\alpha }u-c\in \mathrm{ran}{I}_1^{\alpha }$; we prove a Volterra representation for its elements, graph-norm completeness, and that $\mathcal{L}$ is the adjoint of its zero-trace restriction. Four bounded endpoint traces give an ordinary boundary triplet, so all self-adjoint extensions correspond to Lagrangian planes in ${\mathbb{C}}^4$, including coupled endpoint conditions. For each we give an exact zero-eigenvalue criterion and, when $0\in \rho ({\mathcal{L}}_{\omega })$, an inverse of the form $I_0^{\alpha }{I}_1^{\alpha }$ plus an explicit finite-rank correction. Sharp singular-value asymptotics $s_n({\mathcal{L}}_{\omega }^{-1})=(\pi n{)}^{-2\alpha }(1+O(n^{-1}))$ yield ${\mathcal{L}}_{\omega }^{-1}\in {\mathfrak{S}}_p⟺p>1\mathrm{/}(2\alpha )$, with weak-Schatten endpoint and Weyl asymptotics — now for \emph{every} invertible boundary plane, not only the positive reference problem. As an application we obtain a mild-solution theorem for a Caputo-time diffusion equation in $C([0,T];L^2(0,1))$.