Symmetric Volume Descent from U4 and a Four-Dimensional Lower Bound

We explain a symmetric deformation that decreases the volume of the four-dimensional constant-width body $U_4$. The emphasis is on why the deformation works, rather than the size of the decrease. Oddness preserves width; convexity requires control of the curvature radii. A positive cap on one support branch is coupled to a negative correction on the adjacent branch. Their first-order volume contributions have different orders in the localization parameter, so the favorable contribution dominates. A small motion towards the ball makes the direction feasible for positive time. In a separate part we give a spectral lower bound for all four-dimensional bodies of constant width one, including an explicit positive refinement. A short projection argument using a three-dimensional theorem of Jiang gives the stronger lower bound $0.17872203$. A substantial gap between the lower and upper bounds remains.