Proposition: No natural pointwise signature change

For $n\ge 2$, there is no rule assigning to each positive-definite inner product $g$ on an $n$-dimensional vector space a Lorentzian bilinear form on that same space, equivariantly under all linear coordinate changes, using $g$ alone.

Proof. At $g=I$, equivariance forces the assigned form to be invariant under every orthogonal transformation. Reflection of individual coordinates forces off-diagonal entries to vanish, and coordinate permutations force all diagonal entries to agree. Thus the form is $\lambda I$, which is definite or degenerate, not Lorentzian.