Spherical Fixed-Point Reasoners: Adaptive Loop Halting on the Hypersphere

Looped (weight-tied) transformers enable adaptive computation by allocating more iterations to harder inputs. Fixed-Point Reasoners (FPRM) achieve this with a damped fixed-point solver, but operate in an unbounded Euclidean residual space with coupled rescaling and relative-norm halting. We introduce Spherical Fixed-Point Reasoners (SFPRM), which performs the same computation on the unit hypersphere of normalized transformers (nGPT). Input injection, damping, and halting become geodesic operations, yielding boundedness by construction and eliminating coefficient coupling. We prove fixed-point existence, local convergence under an explicit step-size condition, and global convergence under strong injection. On Sudoku-Extreme and Maze-Hard, SFPRM achieves $96.6$% and $87.7$%, respectively, outperforming FPRM and other baselines while allocating test-time compute more effectively. Its converged geometry is also interpretable: token–input angles capture puzzle structure and predict correctness.