Global Reeb stability for transversely oriented codimension-one foliations

By F1 all the countable-choice hypotheses of the local suppliers hold. By F2, $S$ is nonempty, open and closed. Since $M$ is connected, $S=M$. Thus every leaf is compact and diffeomorphic to $L$, and in particular has finite fundamental group. By F3 its holonomy is trivial. The closedness supplier proves its limit argument using finite-dimensional $H_{\dim M-1}$, finite compact barriers and one-sheeted collar graphs, including the orientation-double-cover case; no dimension-three substitution is being used.

Apply F4: saturated product neighborhoods give interval charts on the leaf space and bundle trivializations of the quotient. The transverse coordinate changes are smooth and increasing, so these charts define a smooth oriented one-manifold structure on the compact connected Hausdorff quotient. Its circle identification can be made smooth by following a positive smooth vector field around this compact one-manifold. Thus $q:M\to S^1$ is a smooth locally trivial bundle with leaves as fibres.

Choose a smooth transverse vector field projecting under $dq$ to the unit positive vector field on $S^1$: local product lifts are patched with a finite partition of unity, and rescaled to have that projection. Its flow exists for the whole circuit because $M$ is compact. If $L_0=q^{-1}(0)$, flow for time one gives a diffeomorphism $g:L_0\to L_0$. Flow for $0\le t\le 1$ trivializes the pullback bundle over $[0,1]$; at the endpoints $(y,1)$ is identified with $(g(y),0)$. Therefore $M$ is the mapping torus with quotient action $f=g^{-1}$, and F5 confirms that positive return is $g$. Two choices of projecting vector field are joined by their convex interpolation, which still projects to the unit base field; smooth flow dependence supplies an isotopy between their return maps. A closed transversal is a single curve and does not by itself specify a return map on the entire fibre.

The asserted compactness, common leaf type, trivial holonomy, circle leaf space, fibre bundle and mapping torus description now follow from steps 1.1–3.1, with monodromy the whole-fibre return for the chosen connection. Full AC enters through the closedness supplier's finite-CW and rational-homology inputs; F1 only propagates its consequence ${\mathrm{A}\mathrm{C}}_{\omega }$ and does not assert the converse.