BCR Lemma 12: small-level base on the canonical spaces

Let $\beta\ge1$, $\alpha\ge0$, and $w\in\mathbb N$ satisfy $\alpha w^2\le1$. On the canonical space $\mathcal M_w=\mathrm{bcrLevel2}(\beta,w)$, there exists a chunk system satisfying the full quantitative induction package:

Thus the size window is $[3^w/2,3\cdot3^w/2]$, the escape price is $2\beta3^w$, the expected total is at least $\alpha\beta w^2 3^w$, the chunk count is at least $\lceil\alpha\beta w^2\rceil$, and initial history is constant.

This is the small-level base case of Lemma 12. Formalization note. The paper uses a path for all small levels; the canonical family instead glues at every level. The statement transports the path construction to a marked geodesic in that canonical space, with $\beta+1$ marked positions spaced $3^w$ apart. The per-chunk bounds quantify over every initial evader location, so the first chunk must pin the starting endpoint.