Calibrations for the Sasaki volume on odd spheres and the no-gap problem

For each odd sphere $S^n$ with $n=2m+1\ge 5$, we consider the Sasaki volume functional ${\mathrm{V}\mathrm{o}\mathrm{l}}^S(V)={\int }_{S^n}\sqrt{\det (I+(\mathrm{\nabla }V{)}^{\mathrm{\top }}(\mathrm{\nabla }V))}d\mathrm{v}\mathrm{o}\mathrm{l}$ on smooth unit tangent vector fields $V$. Using the Gluck–Ziller calibration $\omega =a\wedge \mathrm{\Theta }$ on the unit tangent bundle $E=UT{S}^n$ (extended to constant sectional curvature by Brito–Chacón–Naveira), we establish the universal calibrated lower bound ${\mathrm{V}\mathrm{o}\mathrm{l}}^S(V)\ge c(m;1)\mathrm{v}\mathrm{o}\mathrm{l}(S^n)$, where $c(m;1)=4^m\mathrm{/}\left(\genfrac{}{}{0px}{}{2m}{m}\right)$. In the relaxed (integral-current) setting, we show that the section-constrained stable mass in $E$ equals the calibration value and is attained by an $\omega$-calibrated mass-minimizing integral $n$-cycle in the section class. We also analyze the equality case on smooth graphs. If a smooth graph is $\omega$-calibrated on an open set, then it satisfies the rigidity system ${\mathrm{\nabla }}_{V}V=0$ and ${\mathrm{\nabla }}_{X}V=\lambda X$ for all $X\perp V$, hence is locally a radial distance-gradient field. In particular, for $m\ge 2$ there is no smooth unit field on $S^n$ whose graph is $\omega$-calibrated everywhere. Finally, we construct an explicit smooth recovery sequence (presented in detail for $S^5$ and then extended to all odd dimensions) and prove a uniform nonvanishing estimate for the polar-shell normalization in the patching construction. As a consequence, ${\inf }_V{\mathrm{V}\mathrm{o}\mathrm{l}}^S(V)=c(m;1)\mathrm{v}\mathrm{o}\mathrm{l}(S^n)$, so there is no Lavrentiev gap.