Remark: A fiber metric is not the pairing

The constant metric $\eta$ induces an isomorphism ${\mathrm{Sym}}^2{T}^{\ast }M\cong {\mathrm{Sym}}^{2}TM$ and a fiber metric on ${\mathrm{Sym}}^2{T}^{\ast }M$, whose fiber dimension is also ten. Neither is part of C1. The pairing of C1(iv) is the duality pairing between the value bundle $E_{\mathrm{res}}$ and the test bundle ${E_{\mathrm{res}}}^{\ast }\otimes |\Lambda {|}_W$; the coincidence of ranks does not identify the two roles.