The geodesic spray is a well-defined smooth vector field on TM

On the overlap, $w^a=(\mathrm{\partial }{y}^a\mathrm{/}\mathrm{\partial }{x}^i)v^i$. Along a local integral curve of the $x$-formula, differentiation gives ${\dot{y}}^a=\frac{\mathrm{\partial }{y}^a}{\mathrm{\partial }{x}^i}{v}^i=w^a,$ and ${\dot{w}}^c=\frac{{\mathrm{\partial }}^2{y}^c}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}{v}^i{v}^j-\frac{\mathrm{\partial }{y}^c}{\mathrm{\partial }{x}^k}{\mathrm{\Gamma }}^k{}_{ij}{v}^i{v}^j\mathrm{.}$ Differentiating the inverse-coordinate identity twice gives $\frac{{\mathrm{\partial }}^2{y}^c}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}+\frac{\mathrm{\partial }{y}^c}{\mathrm{\partial }{x}^k}\frac{{\mathrm{\partial }}^2{x}^k}{\mathrm{\partial }{y}^a\mathrm{\partial }{y}^b}\frac{\mathrm{\partial }{y}^a}{\mathrm{\partial }{x}^i}\frac{\mathrm{\partial }{y}^b}{\mathrm{\partial }{x}^j}=0.$ Inserting this and [F2] yields ${\dot{w}}^c=-{\tilde{\mathrm{\Gamma }}}^c{}_{ab}{w}^a{w}^b$, exactly the $y$-formula.

Step 1.1 is the tangent-coordinate transformation law for the local vector fields, so the formulas glue to one vector field on $TM$. Their coordinate components are smooth by [F1], hence the glued field is smooth.

If $(x(t),v(t))$ is an integral curve, its first component equation says $v^i={\dot{x}}^i$ and its second says ${\ddot{x}}^k+{\mathrm{\Gamma }}^k{}_{ij}{\dot{x}}^i{\dot{x}}^j=0$; [F3] therefore makes $x(t)$ a geodesic and $(x,v)$ its velocity lift. Conversely, a geodesic and its velocity satisfy those two equations by [F3], so its lift is an integral curve. At $v=0$ the lift is stationary; dimensions zero and one reduce respectively to the empty system and the scalar calculation, and the empty bundle is harmless. Parameter endpoints are local and one-sided where included. No choice occurs in the overlap calculation; ${\mathrm{A}\mathrm{C}}_{\omega }$ remains the explicit hypothesis inherited from [F1].