Path-space base chunk system with a free escape price

The base case of the BCR chunk induction, with the escape price as a free parameter. On the path metric on $\{0, 1, \dots, \beta\}$ (consecutive points at distance $1$), the marked endpoints are at distance $\beta$, and for any $\alpha \ge 0$ and level $w$ with $\alpha w^2 \le 1$, and any escape price $p \ge 1$, there is a chunk system between the endpoints with deterministic outcomes, sizes in $[\tfrac12, \tfrac32]$, at least $\lceil \alpha\beta w^2 \rceil$ chunks, expected total at least $\alpha w^2 \beta$, and per-chunk conditional cost claims financed by the escape price $p$. The cruel single-file request pattern forces cost $1$ per chunk however the evader plays, and bailing at any prefix costs the full price $p \ge 1$, so any price of at least one unit is sound. Freeing the price from the previously fixed value $2\beta$ lets the level recursion carry a price on the chunk-size scale, as required by the coin-phase separation arithmetic of the race construction.