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Let 
$q$ be an anisotropic quadratic form defined over a general field 
$F$. In this article, we formulate a new upper bound for the isotropy index of 
$q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where 
$\text{char}(F)\neq 2$, and (ii) the case where 
$\text{char}(F)=2$ and 
$q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.
