Remark: Only conditional conformal information

If a later extraction supplies the hypotheses of Theorem thm:classification, the selected cone determines the Lorentzian quadratic form

up to positive multiplication. This concerns the full cone, not merely the comparisons among a sparse set of embedded events CheungEn08A (prop:quadraticunique).

For completeness, the elementary uniqueness calculation is as follows. In Lorentz coordinates write another quadratic form with the same null boundary and positive sign on the cone interior as

Its vanishing on $(1,u)$, $\Vert u{\Vert }_2=1$, and on $(1,-u)$ gives $b=0$ and $u^{\mathsf{T}}Cu=-a$ for every unit $u$. Thus $C=-aI$, and positivity on $(1,0)$ gives $a>0$.

The ambiguity here is a positive constant on one affine carrier. A future field of cones on a manifold would require separate arguments and could leave a position-dependent conformal factor. No metric field or manifold is constructed in XXII-A.