The actual supported Euler class has the generic length coordinate

Smoothness and finite type over the field make $U$ regular, Noetherian and finite-dimensional, and $C$ is an integral Noetherian scheme of dimension one. Apply [F1] to this ambient $U$ and its closed subscheme $C$. Apply generic rank [F2] to the resulting class in $G_0(C)$. Its formula is precisely ${\chi }_{\eta }(E)$, with the cohomological Euler sign $(-1{)}^n$. Finiteness of both the ideal filtration and cohomological sum is part of [F1]–[F2]. The composition is a homomorphism on the actual spectral degree-zero group, and is therefore additive under exact triangles.

Restriction to an open containing $\eta$ leaves the stalk, its local ring and every quotient length unchanged. For $i_{\ast }{\mathcal{O}}_C[0]$ the sole nonzero generic cohomology module is the residue field $k(\eta )$ of ${\mathcal{O}}_{U,\eta }$, which has length one; its cohomological degree is zero. A closed-point-supported object has zero stalk at $\eta$. This proves the positive and zero test values and restriction invariance.

Use [F3] to transport these statements through the actual support-category-to-fiber equivalence. It respects support inclusion and open restriction, so no extra sign or choice of K-spectrum identification is inserted by writing the same class in the open-complement fiber. This proves the stated model qualification.