Curvature as a control field in helicoidal two-body systems

Geometry is increasingly recognized as an active physical resource capable of modifying the behavior of quantum systems beyond conventional external control mechanisms. Here, we establish a curvature--gauge framework in which the geometry of an embedded manifold functions as a tunable control field for classical trajectories and quantum states. By deriving an exact reduced Hamiltonian for a two-body system confined to a helicoidal surface, we demonstrate that curvature modifies the effective kinetic structure while a projected gauge field reconstructs the conserved momentum landscape. This geometric renormalization produces controllable transitions between distinct dynamical regimes, including bounded phase-space structures, zero-energy localization, symmetry-breaking bifurcations, and critical soft-mode behavior. Quantization of the geometry-dependent effective potential reveals a corresponding spectral reconstruction, where harmonic confinement evolves into quartic critical states at the localization threshold. These results establish engineered geometry as a general route for controlling localization and spectral organization in curved quantum, electronic, photonic, and synthetic platforms, where deformation itself becomes a functional degree of freedom rather than a passive constraint.