I am reading about the Thurston norm out of Candel and Conlon's "Foliations 2" and Calegari's "Foliations and the Geometry of 3-Manifolds".

I'm trying to work through the proof of the "Fibered Faces" Theorem. I will denote the Thurston norm by $\xi$ throughout this post.

I understand Calegari's **Lemma 5.14**: if you have a "seed fibration" (that is, a fibration of $M$ over $S^1$ with surface $S$ and $\chi(S) < 0$), it follows that $\xi(\bullet) = |e(T\mathcal{F}) \cap \bullet \ |$ holds on a neighborhood of $[S]$ in $H_2(M; \mathbb{R})$. This proof essentially relies on the fact that the Euler class pairing $\langle e(T\mathcal{F}), \bullet \rangle$ spits out an integer, so it must be constant in a small $\epsilon$-ball around the seed fibration in $H_2(M; \mathbb{R})$.

However, the next theorem (Calegari's **Theorem 5.15**, which is the Fibered Faces Theorem), seems to imply that this pairing $\langle e(T\mathcal{F}), \bullet \rangle$ is constant on a fibered face: he states that given a fibered ray with associated foliation $\mathcal{F}$, and $S$ some Thurston norm minimizing surface on a ray intersecting that same face $F$, $$e(T\mathcal{F}) \cap [S] = \xi([S]) = -\chi(S)$$ The assumption that $S$ is in the $\epsilon$-ball doesn't arise.

Why does this equality hold? Is $\langle e(T\mathcal{F}), \bullet \rangle$ actually constant on this face?

I am trying to reconcile this statement with the one in Candel and Conlon, since they are very careful to specify that the Euler class is constant in a cone of the $\epsilon$-ball.

Any insights are sincerely appreciated!