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

We study the one-parameter family of Riemannian metrics g_λ induced by the Hamiltonian cotangent deformation H_λ= H₀ + λ2\|J\|² on simply connected space forms of constant curvature κ∈ \1, 0, -1\, where J is the momentum map of the isotropy action fixing a base point. The master reciprocal identity 1/f_λ² = 1/s_κ² + λ governing the warping function controls the geometry of the family and implies that radial geodesic acceleration is independent of $λ$. results: (1) Curvature and rigidity: closed-form sectional curvatures, the sharp positivity trichotomy, pointwise and global Brendle-Schoen $1/4$-pinching windows on Sⁿ⁺¹, never-Einstein rigidity for λ≠ 0, and the realization of Hamilton's steady cigar soliton on (H², g₁) as the unique gradient Ricci soliton in the family. (2) Dynamics and stability: iso-kinematic decoupling into unperturbed radial motion and uniform precession $Ω= λL$, closed-form Clairaut integration, the transcendental inward conjugate point law with Euclidean inversion r₀ R^* = 1/λ, and stratified A₃ cusp caustics. (3) Integrability and spectral theory: maximal superintegrable rational pericenter invariants on Rⁿ⁺¹, transverse Killing rigidity on curved space forms, Fuchsian Heun spectral reduction with modulus $τ= 1 + λ/κ$, and essential spectrum gap collapse to [0, ∞) on Hⁿ⁺¹. Extensions to anisotropic sphere families and intrinsic deformations of compact rank-one symmetric spaces (CPⁿ, HPⁿ, OP²) are also established.