The Geometry of the First Moment: A Carrollian Limit

The provided text examines a counter-rotating elliptic universe where the initial state functions as a Carrollian manifold with an effective speed of light derived from the velocity of time. The sources establish that this model's kinematics is identical to Carrollian gravity, utilizing the Hopf fibration to treat the temporal direction as circular fibers over a spherical base. Several mathematical features emerge, including a clock helicity corresponding to the Euler number of the bundle, a closed-orbit obstruction that restricts certain null string theories, and the collapse of the geometric trinity of gravity at equilibrium. Ultimately, the framework reinterprets the universe's expansion and time as internal measurements registered by observers within an atemporal congruence of straight lines in Euclidean four-space.