A temporal covector has uniform compact negativity and orientation margins

Use the smooth charts and the tensor-coordinate trivializations from F6. In a chart $(U,x)$ restricted to $O$, write $e_a=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a$ and let $d{x}^a$ be its dual basis. Then $N={\sum }_a{N}^a{e}_a$, $\alpha ={\sum }_a{\alpha }_{a}d{x}^a$ and $g={\sum }_{a,b}{g}_{ab}d{x}^a\otimes d{x}^b$, with $N^a=d{x}^a(N)$, ${\alpha }_a=\alpha (e_a)$ and $g_{ab}=g(e_a,e_b)$. For any of these bundles, a section $s$ has trivialization coordinates $p\mapsto (p,c(p))$, where $c$ is precisely its finite list of field components. A trivialization and its inverse are continuous, so $s$ is continuous on the chart domain exactly when $c$ is continuous: the forward direction projects the continuous map $(p,c(p))$ to its fibre coordinates, and the reverse direction composes this product map with the inverse trivialization. In total-space smooth coordinates the same representative is $x(p)\mapsto (x(p),c(p))$, so F6's coordinate definition of $C^2$ gives the corresponding equivalence for $C^2$ sections. Changing a bundle chart replaces $c$ by a smooth matrix times $c$ and changes the base coordinates smoothly; hence continuity and $C^2$ regularity are preserved, the latter by F6's algebra and composition rule. Consequently $N^a$ and ${\alpha }_a$ are continuous and $g_{ab}$ is $C^2$ in every smooth chart on $O$, at exactly the regularities stated.

At $p\in O$, let $G=(g_{ab}(p))$ in a smooth coordinate chart. Nondegeneracy gives an inverse matrix $(g^{ab}(p))$. Define $v^a={\sum }_b{g}^{ab}(p){\alpha }_b(p)$. Matrix multiplication gives $g_p(v,w)={\alpha }_p(w)$ for every $w\in T_{p}O$. This vector is unique, since a difference of two candidates pairs to zero with every $w$ and $g_p$ is nondegenerate. Thus the coordinate construction is independent of the chart and defines ${\alpha }^{\mathrm{♯}}$. The same construction for any covector $\beta$ defines $g^{-1}(\alpha ,\beta )$ with coefficients $g^{ab}$ and gives $g({\alpha }^{\mathrm{♯}},{\alpha }^{\mathrm{♯}})=g^{-1}(\alpha ,\alpha )$ and $g({\alpha }^{\mathrm{♯}},N)=\alpha (N)$. By step 1.1 the metric coefficients are $C^2$, so F3 makes the inverse coefficients $C^2$; finite sums and products of them with the continuous field components show that $g^{-1}(\alpha ,\alpha )$, $\alpha (N)$ and $g(N,N)$ are continuous. This constructs the convention of F1 without imposing smoothness on $g,N$ or $\alpha$.

Fix $p\in O$ and a $g_p$-orthonormal basis from F2, and write a timelike vector $v$ and $N_p$ as $(v^0,\mathbf{v})$ and $(n^0,\mathbf{n})$. Their timelikeness gives $\mathrm{\mid }\mathbf{v}\mathrm{\mid }<\mathrm{\mid }{v}^0\mathrm{\mid }$ and $\mathrm{\mid }\mathbf{n}\mathrm{\mid }<\mathrm{\mid }{n}^0\mathrm{\mid }$, so F5 yields $\mathrm{\mid }\mathbf{v}\cdot \mathbf{n}\mathrm{\mid }\le \mathrm{\mid }\mathbf{v}\mathrm{\mid }\mathrm{\mid }\mathbf{n}\mathrm{\mid }<\mathrm{\mid }{v}^0{n}^0\mathrm{\mid }$. Consequently $g_p(v,N_p)=-v^0{n}^0+\mathbf{v}\cdot \mathbf{n}$ is negative exactly when $v^0,n^0$ have the same sign, and positive exactly when their signs differ. F2 identifies these alternatives with the same and opposite timelike components. Therefore, relative to the component selected by $N_p$, a timelike $v$ is past directed exactly when $g_p(v,N_p)>0$. This is a pointwise test and requires no smooth orientation field or smoothly varying orthonormal basis. Applied to $v={\alpha }^{\mathrm{♯}}$, step 2.1 proves the asserted temporal-covector convention, including the past-timelike conclusion wherever the two defining inequalities hold.

Put $h=-g^{-1}(\alpha ,\alpha )$, $u=\alpha (N)$ and $z=-g(N,N)$. These functions are continuous on $O$ and positive on $K$. For each $p\in K$, define $W_p$ to be the set of $q\in O$ satisfying $h(q)>h(p)\mathrm{/}2$, $u(q)>u(p)\mathrm{/}2$ and $z(q)>z(p)\mathrm{/}2$. This is an open neighborhood of $p$. The indexed family $(W_p{)}_{p\in K}$ covers $K$; F4 gives finitely many centers $p_1,\dots ,p_r$ whose neighborhoods cover $K$. Set $m={\min }_{i}h(p_i)\mathrm{/}2$, $a={\min }_{i}u(p_i)\mathrm{/}2$ and $b={\min }_{i}z(p_i)\mathrm{/}2$. The finite list is nonempty and all its entries are positive. These constants give the three required uniform inequalities throughout $K$.

F4 supplies a finite compact chart cover inside $O$, with pieces in $K$ if desired. Now fix any cover allowed in the Statement, not necessarily that particular one. Each component function $f$ of either field is defined and continuous on $O\cap U_j$, which contains $K_j$. If $K_j$ is nonempty, cover it by the open neighborhoods on which $\mathrm{\mid }f(q)\mathrm{\mid }<\mathrm{\mid }f(p)\mathrm{\mid }+1$ for centers $p\in K_j$. A finite subcover from F4 bounds $\mathrm{\mid }f\mathrm{\mid }$ on $K_j$ by the maximum of the finitely many numbers $\mathrm{\mid }f(p)\mathrm{\mid }+1$. For empty pieces the bound is vacuous. Repeating this for the finitely many components in the finitely many specified charts proves the component conclusion. Step 3.1 proves the past-timelike conclusion on $K$, and step 3.2 proves the uniform margins, completing all the claims.