The Mobius core circle has no integral oriented self-intersection but mod two data survives

The quotient has local charts obtained by lifting base intervals of length less than one to $\mathbb{R}$; on overlaps the lifted base coordinates differ by an integer $k$ and the fibre changes by $(-1{)}^k$. These charts are smooth and fibre-linear, so [F1] gives a line bundle over the smooth quotient circle. The chart maps are homeomorphisms because their integer translates have open saturation and are disjoint over each lifted interval. Distinct base points are separated by the Hausdorff quotient circle, and distinct points over one base point are separated in a bundle chart; thus the total space is Hausdorff. Rational-endpoint lifted base intervals and fibre intervals give a countable base. The image of $[0,1]$ covers $Q$, making it compact. Its interval charts have integer-translation transitions, so its zero section is a compact boundaryless embedded circle. Along it $TL{\mathrm{\mid }}_Z=TQ\oplus L$, so the normal quotient is $L$. A two-arc presentation has transition $+1$ on one overlap component and $-1$ on the other; putting $-1$ on both components would instead be a trivial bundle.

Nonorientability. An orientation pulled back to the connected covering $\mathbb{R}$ would be a continuous sign $o(t)\in \{+1,-1\}$, hence constant, but the deck change requires $o(t+1)=-o(t)$, a contradiction. The total-space deck map $(t,v)\mapsto (t+1,-v)$ has determinant $-1$; the same argument on its connected covering plane proves that the total space is nonorientable. Thus [F3] gives $w_1(L)\mathrm{\ne }0$.

For $0<\epsilon <1$ the function $f(t)=\epsilon \sin (\pi t)$ obeys $f(t+1)=-f(t)$ and therefore defines a smooth section of $L$. Its zeros are all integers, which give exactly one point of $Q$; its vertical derivative there is $\epsilon \pi \mathrm{\ne }0$ in a lifted chart. Its graph is a small push-off of the zero section and meets it transversely once, so $Z{\cdot }_{2}Z=1$. By [F4] this equals $⟨w_1(L),[Q]{⟩}_2$, without assuming an unproved cohomology computation or using a rank-one projective fibre-generator claim.

The orientation obstruction in step 2.1 violates [F5], so no untwisted integral oriented self-intersection is defined by that construction. Changing a local fibre trivialization can reverse a local zero sign, and there is no continuous global choice making all such signs consistent. The ambiguity is not merely a single overall sign for an arbitrary finite zero set. Modulo two every local sign is one and step 2.2 gives the invariant count. AC is inherited from [F3]–[F4]; the explicit quotient and section use no extra choice.