The base case: a filtered chunk system on the path

The base case of BCR's Lemma 6, in the filtered form: 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 filtered chunk system with $m = \beta$ chunks: after an initial pinning request $\{0\}$, the $i$-th chunk is the single request $\{i\}$, of size $1 \in [\tfrac12, \tfrac32]$, with trivial filtration. The total size $\beta$ dominates $\alpha w^2\beta$, and $m = \beta \ge \lceil \alpha\beta w^2\rceil$; the conditional cost bound holds with escape price $2\beta$ because the previous singleton request pins the evader one step away.

This is the same statement as chunk_system_base transported to the filtered package KServer_chunk_system_f, which is the form consumed by the combining and subchunk steps of the induction.

Formalization note

The filtration is constant (hist ≡ 0): the system is deterministic.