An Abresch-Rosenberg type tensor and a gap theorem for CMC hypersurfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$
We construct a symmetric trace-free tensor for constant-mean-curvature hypersurfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$ that extends the Abresch-Rosenberg tensor. Within its natural class, it is characterized by the linearity of its Codazzi defect, and its complete connected zero set consists exactly of the rotational CMC spheres. An exact integrated Simons identity yields explicit gap theorems, valid for every $h>1$ in $\mathbb{H}^n\times\mathbb{R}$ and every $h>(n\int_0^{π/2}\sin^{n-1}r\,dr)^{-1}$ in $\mathbb{S}^n\times\mathbb{R}$. We compare the pinching thresholds with the pointwise algebraic level that implies convexity. In $\mathbb{H}^n\times\mathbb{R}$ the explicit pinching conditions imply strict convexity, so there the method gives an independent proof with explicit constants. In $\mathbb{S}^n\times\mathbb{R}$ the pinching region exceeds the convexity level for $n\in\{3,4\}$ and every admissible $h$, and for $n\geq5$ whenever $h^2\geq n/(6(n+2))$; there the gap theorem does not require convexity to be known in advance. We also determine second-order large-$h$ asymptotics of the thresholds.