Counterexample: a future null curve has no proper-time parametrization

In an inertial chart with ${\eta }_0=\mathrm{diag}(-1,1,1,1)$ let $k:=(1,1,0,0)$: then $\eta (k,k)=-1+1=0$ and $k\mathrm{\ne }0$, so $k$ is null by [L1], and its first coordinate is positive, so $k$ lies in the closure of the future timelike component and is future null by [L2].

The curve $x(s):=p+sk$ has difference quotients $(x(s+h)-x(s))\mathrm{/}h=k$ for every $h\mathrm{\ne }0$, so $x^{\mathrm{\prime }}(s)=k$ for every $s$ and $x$ is a regular $C^1$ future null curve by [L4].

Evaluating the length density appearing in [L5] gives $\sqrt{-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))}=\sqrt{-\eta (k,k)}=0$ identically; hence the formal length integral $L(s):=c^{-1}{\int }_0^s\sqrt{-\eta (k,k)}dr=0$ is constant and cannot be inverted to give a parameter. The proper-time function of [L5] is not defined on this null curve; the failure of its formal extension refutes the proposed universal construction.

No normalized four-velocity can be attached to the curve either: the tangent is $k$, and every scalar multiple $\lambda k$ has squared norm $\eta (\lambda k,\lambda k)={\lambda }^2\eta (k,k)=0$ by bilinearity, while a four-velocity $U$ with $\eta (U,U)=-c^2<0$ would be timelike, and a vector cannot be both null and timelike by [L1]. In the chart its coordinates are $x_A(p)+(s,s,0,0)$; with coordinate time $t=(x_A^0(p)+s)\mathrm{/}c$, the spatial coordinate velocity is $(c,0,0)$. This is a geometric speed computation, not a physical light-ray identification.