A strict gap above the Brito--Chacon--Naveira bound for the Sasaki volume on higher odd spheres

For a smooth unit vector field $V$ on the round sphere $S^{2m+1}$, its Sasaki volume is the volume of its graph in the unit tangent bundle. Brito, Chacón and Naveira proved that this volume is at least $c_m\operatorname{vol}(S^{2m+1})$, where $c_m=4^m/\binom{2m}{m}$. We prove that, for every $m\ge2$, the infimum over smooth unit fields is strictly larger than this value. The proof constructs a global closed comass-one form on $T^1S^{2m+1}$ and classifies all of its equality planes. A hypothetical sequence approaching the bound has a calibrated integral-current limit. Its positive horizontal-Jacobian part is a multiplicity-one graph, whereas its projection-degenerate residual is confined to a base-rank-one equality plane. Coordinate test forms show that this residual contributes no singular part to the distributional derivative of the graph section. The section is therefore Sobolev and satisfies a scalar concircular equation; a weak rigidity theorem identifies it with a radial distance-gradient field. The radial graph has the two pole fibers as boundary, while a local boundary--mass identity forces the residual to vanish, contradicting the cycle condition. We also prove a sharp current-level flat-boundary inequality for fillings of the pole fibers. Consequently, every sequence of smooth unit fields converging in measure to a radial field has lower limiting volume at least twice the Brito--Chacón--Naveira bound. The strict gap obtained here is qualitative; no explicit gap constant is claimed.