Robust Non-Singular Bouncing Cosmology from Regularized Hyperbolic Field Space

We present a framework for non-singular bouncing cosmology in a closed ($k=+1$) universe with a two-field sigma model whose regularized hyperbolic field-space metric $g_{\chi \chi }^S(\phi )=(1+e^{-2\alpha \phi \mathrm{/}{M}_{\mathrm{P}\mathrm{l}}}{)}^{-1}$ is derived from three physical boundary conditions: (i) kinetic suppression during contraction enabling the bounce, (ii) canonical normalization during inflation preserving perturbative unitarity, and (iii) positive-definiteness ensuring ghost-freedom. The bounce preserves the Null Energy Condition and is BKL-stable. The full two-field perturbation system $(\delta \phi ,\delta \chi ,\mathrm{\Phi })$ is integrated in the Newtonian gauge through the bounce over 65 e-folds, circumventing the comoving-gauge $H=0$ singularity, with both Einstein constraints verified a posteriori. Scalar sound speeds numerically measured adjacent to $H=0$ satisfy $\mathrm{\mid }{c}_{\phi }^2-1\mathrm{\mid },\mathrm{\mid }{c}_{\chi }^2-1\mathrm{\mid }\le 8\times {10}^{-16}$, establishing strict hyperbolicity (no ghost or gradient instability); $\mathcal{R}$ is conserved on super-Hubble scales to $\mathrm{\mid }\mathrm{\Delta }{\mathcal{R}}^2\mathrm{/}{\mathcal{R}}^2\mathrm{\mid }=4.43\times {10}^{-3}$. An independent CMB-scale Mukhanov-Sasaki integration confirms $n_s=0.9683$, matching exact Starobinsky slow-roll to $\mathrm{\mid }\mathrm{\Delta }{n}_s\mathrm{\mid }=5.47\times {10}^{-4}$. $\delta N$ non-Gaussianity yields $f_{\mathrm{N}\mathrm{L}}^{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}=+0.0133$, consistent with Maldacena’s relation to $\mathrm{\mid }\mathrm{\Delta }{f}_{\mathrm{N}\mathrm{L}}\mathrm{\mid }=1.54\times {10}^{-4}$. A bounce-scale spectral feature is pushed by $\sim 2630$ post-bounce e-folds to $k_{\mathrm{C}\mathrm{M}\mathrm{B}}\mathrm{/}{k}_H\sim {10}^{1116}$, far beyond the observable universe, so the model recovers Starobinsky predictions on all observable scales while resolving the initial singularity. Predictions ($n_s\approx 0.967$, $r\approx 0.003$, $f_{\mathrm{N}\mathrm{L}}^{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}\approx +0.013$, independent of $\alpha$) are consistent with Planck 2018 and testable by next-generation CMB experiments.