The base case: a chunk system on the path

The base case of BCR's Lemma 6: on the path of $\beta + 1$ equally spaced points (with $s = 0$, $t = \beta$, so $d(s,t) = \beta$), whenever $\alpha w^2 \le 1$ there is a (deterministic) chunk system with $m = \beta$ chunks: after an initial pinning request $\{0\}$, the $i$-th chunk is the single request $\{i\}$, of size $c_i = 1 \in [\tfrac12, \tfrac32]$. The total size $\beta$ dominates $\alpha w^2 \beta$, and $m = \beta \ge \lceil \alpha\beta w^2 \rceil$.

The conditional cost bound holds against every evader with escape price $2\beta$: after serving the requests $\{0\}, \dots, \{i-1\}$ the evader's position is pinned at $i - 1$, so serving $\{i\}$ costs exactly $1$, while bailing out costs $2\beta \ge 1$; the first chunk costs at least $1$ from any starting position because it pins the evader at $0$ before requesting $\{1\}$. The offline evader serves everything for cost $\beta$ by walking down the path.

Role

This is the induction base of the Bubeck–Coester–Rabani $\Omega(\log^2 k)$ lower bound (STOC 2023, Section 4): all levels $w$ with $\alpha w^2 \le 1$ use this path system; the six-copy cyclic construction then builds the higher levels.

Formalization note

The path metric is induced from the embedding into ℝ. The first chunk is $[\{0\}, \{1\}]$ rather than $[\{1\}]$, pinning the start (the evader model has no distinguished starting point); the sequence convention "first request $\{s\}$" matches BCR.