Affine generator profiles give constant signed acceleration

Fix $p\in H$. By [F3], shrink a slice chart to a product coordinate box with coordinates $(x^1,x^2,x^3,r)$ and $H=\{r=0\}$ in this box; put $x^4=r$. The smooth coordinate coefficients of fields along $H$, and of functions on $H$, extend smoothly by making them independent of $r$. Thus the local ambient extensions in the Statement exist. Define smooth functions ${\mathrm{\Gamma }}^a{}_{ij}$ in this chart by ${\mathrm{\nabla }}_{{\mathrm{\partial }}_i}{\mathrm{\partial }}_j={\mathrm{\Gamma }}^a{}_{ij}{\mathrm{\partial }}_a$. Expanding the ambient connection rules in [F1] gives $({\mathrm{\nabla }}_{\tilde{X}}\tilde{Y}{)}^a={\tilde{X}}^i{\mathrm{\partial }}_i{\tilde{Y}}^a+{\mathrm{\Gamma }}^a{}_{ij}{\tilde{X}}^i{\tilde{Y}}^j$, with indices from $1$ to $4$.

If two extensions of $Y$ differ by $A$, then $A^a(x^1,x^2,x^3,0)=0$, so ${\mathrm{\partial }}_i{A}^a=0$ on $H$ for $i=1,2,3$. Since $X$ is tangent, ${\tilde{X}}^4{\mathrm{\mid }}_H=0$. The formula in step 1.1 therefore gives $({\mathrm{\nabla }}_{\tilde{X}}A){\mathrm{\mid }}_H=0$. Changing an extension of $X$ leaves its values on $H$ fixed, so the same formula gives no change in the derivative there either. The local definitions agree on overlaps because each equals restriction of the same intrinsic ambient connection applied to any local extensions. They consequently define a smooth field along $H$, independent of chart and extensions. Extending a scalar $f$ in the same way gives $\tilde{X}(\tilde{f}){\mathrm{\mid }}_H=X(f)$ by [F3]; restricting the ambient rules therefore proves ${\mathrm{\nabla }}_{fX}Y=f{\mathrm{\nabla }}_{X}Y$ and ${\mathrm{\nabla }}_X(fY)=X(f)Y+f{\mathrm{\nabla }}_{X}Y$ along $H$.

Using the now well-defined restricted derivative and its rules, ${\mathrm{\nabla }}_{csL}(csL)=cs\{cL(s)L+cs{\mathrm{\nabla }}_{L}L\}=c^{2}sL=cK$. This calculation is valid everywhere on $H$, including $s=0$, using the given $L(s)=1$ and affine condition ${\mathrm{\nabla }}_{L}L=0$.

Replacing $c$ by $-c$ gives the coefficient $-c$ for the other branch. Off the section the coefficients are unique because $K\mathrm{\ne }0$ by [F2]. On the section, $K=0$ makes both sides zero for any assigned scalar; the constant branch values are limits of the values off the section, not values deduced from this zero equation.