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 $\mathrm{Vol}^S(V)=\int_{S^n}\sqrt{\det(I+(\nabla V)^\top(\nabla V))}\,d\mathrm{vol}$ on smooth unit tangent vector fields $V$. Using the Gluck--Ziller calibration $\omega=a\wedge\Theta$ on the unit tangent bundle $E=UT S^n$ (extended to constant sectional curvature by Brito--Chac\'on--Naveira), we establish the universal calibrated lower bound $\mathrm{Vol}^S(V)\ge c(m;1)\,\mathrm{vol}(S^n)$, where $c(m;1)=4^m/\binom{2m}{m}$. 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 $\nabla_V V=0$ and $\nabla_X V=\lambda X$ 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, $\inf_V \mathrm{Vol}^S(V)=c(m;1)\,\mathrm{vol}(S^n)$, so there is no Lavrentiev gap.