Hamiltonian Cotangent Deformations of Space Forms: Curvature Pinching, Soliton Rigidity, Conjugate Loci, and Superintegrability

We study the one-parameter family of Riemannian metrics $g_{\lambda }$ induced by the Hamiltonian cotangent deformation $H_{\lambda }=H_0+\frac{\lambda }{2}\Vert J{\Vert }^2$ on simply connected space forms of constant curvature $\kappa \in \{1,0,-1\}$, where $J$ is the momentum map of the isotropy action fixing a base point. The master reciprocal identity $1/f_{\lambda }^2=1/s_{\kappa }^2+\lambda$ governing the warping function controls the geometry of the family and implies that radial geodesic acceleration is independent of $\lambda$. From this mechanism we derive three classes of results: (1) Curvature and rigidity: closed-form sectional curvatures, the sharp positivity trichotomy, pointwise and global Brendle-Schoen $1/4$-pinching windows on ${\mathbb{S}}^{n+1}$, never-Einstein rigidity for $\lambda \ne 0$, and the realization of Hamilton's steady cigar soliton on $({\mathbb{H}}^2,g_1)$ as the unique gradient Ricci soliton in the family. (2) Dynamics and stability: iso-kinematic decoupling into unperturbed radial motion and uniform precession $\Omega =\lambda L$, closed-form Clairaut integration, the transcendental inward conjugate point law with Euclidean inversion $r_0{R}^{\ast }=1/\lambda$, and stratified $A_3$ cusp caustics. (3) Integrability and spectral theory: maximal superintegrable rational pericenter invariants on ${\mathbb{R}}^{n+1}$, transverse Killing rigidity on curved space forms, Fuchsian Heun spectral reduction with modulus $\tau =1+\lambda /\kappa$, and essential spectrum gap collapse to $[0,\infty )$ on ${\mathbb{H}}^{n+1}$. Extensions to anisotropic sphere families and intrinsic deformations of compact rank-one symmetric spaces ($\mathbb{C}{P}^n,\mathbb{H}{P}^n,\mathbb{O}{P}^2$) are also established.