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

Abstract

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-\(\varepsilon\) target $R_{\mathrm{ap},\varepsilon}(\mathcal K_F)$ is separated from the closed fixed-\(\varepsilon\) target $C_{\mathrm{ap},\varepsilon}(\mathcal K_F)$ and from the exact approximate point spectrum $\sigma_{\mathrm{ap}}(\mathcal K_F).$ This endpoint statement uses the same point-evaluation plus fixed-quadrature information model as the \(1<p<\infty\) residual theory. Second, we isolate two obstructions at the nonseparable endpoint \(L^\infty\). Fixed quadrature schemes do not discretize the full \(L^\infty\) unit sphere, and even inside measure-preserving Cantor homeomorphisms the map $F\mapsto \sigma_{\mathrm{ap}}(\mathcal K_F:L^\infty\to L^\infty)$ 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 \(L^\infty\) approximate point spectrum: for a fixed non-torsion \(z_0\in\mathbb T\), the condition $z_0\in\sigma_{\mathrm{ap}}(\mathcal K_{F}:L^\infty\to L^\infty)$ collapses to a Borel unbounded-period condition. In addition, fixed \(L^\infty\) 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.