Endpoint Koopman Spectral Computation: $L^1$ Residual Bounds, $L^{\mathrm{\infty }}$ Instability, and Point-Spectral SCI Calibration Families

We study endpoint Koopman spectral computation from the viewpoint of the Solvability Complexity Index (SCI). Let $(\mathcal{X},d)$ be a compact metric space with finite Borel measure $\omega$, and let ${\mathcal{K}}_F$ be the Koopman operator associated with a continuous nonsingular map $F:\mathcal{X}\to \mathcal{X}$. First, on $L^1(\mathcal{X},\omega )$, we record the endpoint residual upper-bound in the target-split form. The regularized compact fixed-$\epsilon$ target $R_{\mathrm{a}\mathrm{p},\epsilon }({\mathcal{K}}_F)$ is separated from the closed fixed-$\epsilon$ target $C_{\mathrm{a}\mathrm{p},\epsilon }({\mathcal{K}}_F)$ and from the exact approximate point spectrum ${\sigma }_{\mathrm{a}\mathrm{p}}({\mathcal{K}}_F)\mathrm{.}$ This endpoint statement uses the same point-evaluation plus fixed-quadrature information model as the $1<p<\mathrm{\infty }$ residual theory. Second, we isolate two obstructions at the nonseparable endpoint $L^{\mathrm{\infty }}$. Fixed quadrature schemes do not discretize the full $L^{\mathrm{\infty }}$ unit sphere, and even inside measure-preserving Cantor homeomorphisms the map $F\mapsto {\sigma }_{\mathrm{a}\mathrm{p}}({\mathcal{K}}_F:L^{\mathrm{\infty }}\to L^{\mathrm{\infty }})$ is maximally discontinuous in Hausdorff distance under arbitrarily small uniform perturbations of $F$. We also show that finite-period Silver-tree block constructions cannot yield analytic hardness for the $L^{\mathrm{\infty }}$ approximate point spectrum: for a fixed non-torsion $z_0\in \mathbb{T}$, the condition $z_0\in {\sigma }_{\mathrm{a}\mathrm{p}}({\mathcal{K}}_F:L^{\mathrm{\infty }}\to L^{\mathrm{\infty }})$ collapses to a Borel unbounded-period condition. In addition, fixed $L^{\mathrm{\infty }}$ point-eigenvalue membership is Borel in the measure-preserving continuous class, so one fixed eigenvalue cannot encode a non-Borel tree predicate. Third, we construct Koopman point-spectrum calibration families on the Cantor space.