We study perturbations through a locally non-singular bounce in a closed universe (\(K=+1\)) with a regularised two-field sigma model. Positive curvature drives the bounce in General Relativity without violating the Null Energy Condition, and the expanding branch settles onto Starobinsky inflation with the standard slow-roll observables at \(N=60\) (\(n_s \approx 0.967\), \(r \approx 0.003\), \(f_{\rm NL} \approx +0.013\)). The scalar sector is integrated through \(H=0\) in two field-space representations of the comoving curvature perturbation \(\mathcal{R}\), both in Newtonian gauge: across 64 turning cells the cross-representation residual stays below \(6\times10^{-10}\), and Einstein-constraint residuals stay at the \(10^{-8}\)–\(10^{-12}\) level across the bounce. On the fiducial no-turn trajectory we derive the exact finite-harmonic closed-\(S^3\) reduced scalar action, propagate two canonically normalised finite-time states, and obtain an early-plateau tensor-to-scalar ratio \(r_{\ell=60} = 1.73\times10^{-6}\) and a direct entropy-to-curvature power ratio of \(0.98\); a closed-FRW tensor solver reproduces the flat slow-roll amplitude to \(0.003\%\). The result is a finite-grid continuation of linear perturbations through a curvature-driven bounce, not a proof of past geodesic completeness.